Spectral Graph Theory (CBMS Regional Conference Series in Mathematics, No. 92)

k-1 L(l - ,\i )ta;b,. i=1

From Lemma 3.9, we know that we can choose two of the subsets, say, X and Y with their associated vectors satisfying

k -1 L(l - ,\i)ta;b; � 0 . i= 1

Therefore, we have volX volY

----,-....,- - ( 1 vol G

-

,

Ak

)t v'volX volX volY volY vol G

and Theorem 3. 10 is proved. We note that the condition 1 as in proof of Theorem 3 . 1 :

0

- ,\k

�

An-!

-

1 can be eliminated by modifyin g

Ak

3.

48

DIAMETERS AND EIGENVALUES

THEOREM 3 . 1 1 . For Xi

C

r

1

V(G) , i = 0, 1, · · · , k, we have

mm d(Xi , Xj ) $ max .

• #J

#;

log log

voiX; volXj voiX; voiX; .>. _ 1 Hr. "

).,. _ , - >.. .

Another useful generalization of Theorem 3.10 is the following:

THEOREM 3 . 1 2 . For Xi

C

V (G) , i

=

r

1

0, 1 , · · · , k, we have

volXj log voiX; voiX; voiX; . mm . d ( Xi , X ) < mm max >.. • - i O �j < k i# j 3 ih log �A nn -- jj -- 1t + - ). lc - j ·

-

where j satisfies Ak-i f. An-j- 1 ·

PROOF . For each j , 1 $ j $ k - 1 , we can use a very similar proof to that in Theorem 3.10, such that there are two of the subsets X and Y with their corre­ sponding vectors satisfying > "'(1 L...J - >.. · )1 a · b · 0 iES

where S

=

{ i : 1 $ i $ k - j or n

-

'

,

, _

j+1 $i $

n

-

1} .

The proof then follows.

0

3.4. Eigenvalue upper bounds for manifolds There are many similarities between the Laplace operator on compact Riemann­ ian manifolds and the Laplacian for finite graphs. While the Laplace operator for a manifold is generated by the Riemannian metric, for a graph it comes from the ad­ jacency relation. Sometimes it is possible to treat both the continuous and discrete cases by a universal approach. The general setting is as follows: 1 . an underlying space M with a finite measure

JL;

2. a well-defined Laplace operator C on functions on M so that C is a self­

adjoint operator in L2 (M, JL) with a discrete spectrum; 3. if M has a boundary then the boundary condition should be chosen so that it does not disrupt self-adjointness of C; 4. a distance function dist (x, y) o n M s o that I Vdist I $ 1 for an appropriate notion of gradient.

For a finite connected graph (also denoted by M in this section) , the metric JL can be defined to be the degree of each vertex. Together with the Laplacian £ , all the above properties are satisfied. In addition, we can consider an r-neighborhood of the support suppr f of a function f in L 2 (M, JL) for r E lR: suppr f = {x E M : dist(x, supp f ) $ r } the distance function in M . For a polynomial of degree where dist denotes denoted by Ps , then we have (3 .5)

supp p, (C)f

C

supp.f.

s,

3.4.

EIGENVALUE UPPER BOUNDS FOR M ANIFOLDS

49

Let M be a complete Riemannian manifold with finite vol ume and let .C be the self-adjoint operator - �, where � is the Laplace operator associated with the Riemannian metric on M (which will be defined later in ( 3.9} , also see [251] } . Or, we could consider a compact Riemannian manifold M with boundary and l et .C be a self-adjoint operator - � subject to the Neumann or Dirichlet boundary conditions (defined in ( 3. 10}) . We can sti ll have the following analogous version of (3.5} for the s-neighborhood of the support of a function. There exists a non-trivial family of bounded continuous functions P, (A ) defined on the spectrum Spec.C, where s ranges over [0, +oo) , so that for any function 2 I E L (M, J.t) :

suppP, (.C)/

( 3.6}

For example, we can choose P, (A) quirement in (3.6) . Let us define

p(s)

==

C

supps f .

==

cos(v'Xs) which clearly satisfies the re­

sup

.XESpec£

I P, ( A ) I

and assume that p( s ) is locally integrable. We consider � (A) =

l)O ¢(s)P, (A}ds

where ¢ ( s} be a measurable function on ( 0, +oo) such that

100 l 4>( s) I p(s) ds < oo .

In particular, � (A) is a bounded function on Spec.C, and we can apply the operator �(.C) to any function in L2 ( M , p). We will prove the following general lemma which will be useful later. LEMMA 3 . 1 3 . If f E L2 ( M , p ) then II � ( .C) / II L2( M\supprf) � 11/ lb

where ll / l l 2 := 1 1 / I I P ( M,�£) · PROOF . Let us denote w( x )

=

� ( .C ) f ( x ) =

1 00 I ¢(s} I p(s}ds

100 c/>(s) P, (.C) f (x) ds.

If the point x is not in supp r f then P, ( .C ) f (x) = 0 whenever s � those points w (x) =

100 ¢(s)P, (.C)f(x) ds

r.

Therefore, for

3. DIAMETERS AND EIGENVALUES

50

and ll w i i £ 2 ( M \supp,!)

< <

::;

The proof is complete.

1 1oo <J> (s)Pa (C) f (x)ds l 2 100 II <J>(s)Pa (C) f(x) ll 2ds 100 I ( s) I p( s) ll f ll 2ds.

0

As an immediate consequence, we have

COROLLARY 3 . 1 4 . If f , g E L 2 ( M, p. ) and the distance between supp f and

supp g is D, then

(3.7)

1/ f � (£)gdp. l ::; ll f ll 2 IIYII 2 koo I <J>(s ) I p(s )ds M

The integral on the left-hand side of (3. 7) is reduced to one over the support of g which in turn is majorized by the integral over the exterior of suppnf- The rest of the proof follows by a straightforward application of the Cauchy-Schwarz inequality. For the choice of P8 ( A ) = cos (\!"Xs) , suppose we select •• 1 ( s) = -e- .. .

v'1rt

Then we have � ( A) =

100 <J>( s) P, (A)ds = e->.t .

CoROLLARY 3 . 1 5 . If f , g E L2 (M, p. ) and the distance between the supports of f and g is equal to D then ( 3.8 )

liM fe - t Cgdp. l ::; ll f ll 2 IIYib Loo � e - ft ds .

Let us mention a similar but weaker inequality:

COROLLARY 3 . 16 .

This inequality was proved in [93) [251) and is quite useful. Let M be a smooth connected compact Riemannian manifold and A be a Laplace operator associated with the Riemannian metric, i.e., in coordinates x 1 , X2 , · . . , X n ,

(3.9)

1 � 8(

..

()u, )

Au = - � - vgg '1 OXj VY i,j = l OX;

3.4.

EIGENVALUE UPPER BOUNDS FOR MANIFOLDS

where gii are the contravariant components of the metric tensor, g

gii = l l9ii 11 - 1 , and u is a smooth function on M.

51

=

det IIYii ll •

If the manifold M has a boundary 8M, we introduce a boundary condition 8u au + /3 a v = 0

(3 . 10)

where a(x), /3 (x) are non-negative smooth functions on M such that a(x) +/3(x) for all x E 8M .

>

0

For example, both Dirichlet and Neumann boundary conditions satisfy these assumptions.

The operator £ = - .1. is self-adjoint and has a discrete spectrum in L 2 (M, J-L ) , where p, denotes the Riemannian measure. Let the eigenvalues be denoted by 0 = Ao < A 1 � A 2 � · • · • Let dist( x , y ) be a distance function on M x M which is Lipschitz and satisfies I Y'dist( x , y) i � 1 for all x, y E M . For example, dist(x, y ) may be taken to be the geodesic distance, but we don't necessarily assume this is the case. We want to show the following (also see [54]):

THEOREM 3 . 1 7 . For two arbitrary measurable disjoint sets X and

(

w e have

)

Y on M,

1 (p,M) 2 2 A1 � 1 + log p,X Y p, dist(X, Y ) 2 X Moreover, if we have k + 1 disjoint subsets X0 , 1 , Xk such that the distance between any pair of them is greater than or equal to D > 0, then we have for an y k � 1, (p,M) 2 1 (3. 12 ) ).k � 2 ( 1 + sup log X X ) 2 . D i#i JJ i JJ i (3. 1 1 )

,

· · ·

PROOF. Let us denote by r/J; the eigenfunction corresponding to the i -th eigen­ value A; and normalized in L 2 (M, J-L) so that {tj>i } is an orthonormal frame in L 2 (M, J-L) . For example, if either the manifold has no boundary or the Dirichlet or Neumann boundary condition is satisfied, there is one eigenvalue 0 with the associated eigenfunction being the constant function: 1 ¢o = -.(jiM .

The proof is based upon two fundamental facts about the heat kernel p(x, y , t) , which by definition is the unique fundamental solution to the heat equation at u ( x , t) -

8

.1.u(x, t) = 0

with the boundary condition (3. 10) if the boundary 8M is non-empty. The first fact is the eigenfunction expansion

(3.13)

p(x, y , t) =

00

:�:::e - -'•t,p; (x)r/J; (y) i =O

52

3.

DIAMETERS AND EIGENVALUES

and the second is the following estimate (by using Corollary 3. 16) : !

fx i p(x, y, t)f(x)g(y)J-'(dx)J-'(dy) 5:(L f2 i g2 ) ex\ - �:)

(3. 14)

for any functions j, g E L2 (M, 1-') and for any two disjoint Borel sets X, Y where D = dist (X, Y ) .

C

M

We first consider the case k = 2. We start by integrating the eigenvalue expan­ sion (3.13) as follows: (3.15)

= r r r lrx Ni }y l x }y i=O We denote by /; the Fourier coefficients of the function f'I/Jx with respect to the frame { ¢1 } and by g1 those of '1/Jy . Then

I ( f , g)

=

p(x,y, t)f(x)g(y)jj(dx)jj(dy) f:e->.;t g e I(f,g) e - >.o t foYo + L i=l ->.' t fi9i

Y¢i·

00

=

where we have used

� � e->., t fi9i l 5: e->.tt (� � ) !l

Y?

I 2

5:

e - >. 1 t ll f'I/Jx ii2 IIY'I/JY II 2 ·

By comparing (3.16) and (3.14) , we have (3. 16) exp( -.>.I ) II f 'I/Jx ll2 l l

'I/J

g y l l2

�

fo go - ll f'I/Jx ll 2 l l g'I/Jy l l 2 exp ( - �: )

·

We will choose t so that the second term on the right-hand side (3.16) is equal to one half of the first one (here we take advantage of the Gaussian exponential since it can be made arbitrarily close to 0 by taking smal l enough) : D2 ,.· 2 - ---:::.,.-:--:--;;-:�..,...-, ll 2 II9.PY II2 11Nx 4 log fo go

t

t-

t

For this we have which implies

After substituting this value of t, we have ,

<

(

.i_ log 2 ll f 'I/Jx ll 2 IIY'I/JY II2 I

AI - D 2

J 090

)

2

Finally, we choose f = g = ¢o and take into account that

fo = L No L ¢�, =

3.4. EIGENVALUE UPPER BOUNDS FOR MANIFOLDS

and

(L ¢>�)

ll/t/Jx l l 2 = Similar identities hold for

1 2

=

53

../To .

We then obtain

g.

(

1 At $ D2

log

2 ) fx ¢>� Jy ¢>� 4

Now we consider the general case k > 2. For a function f ( x) , we denote by the i-th Fourier coefficient of the function / l x; i.e.

J/

J/ = f /¢>; . lx; Similar to the case of k = 2, we have ltm {f, J ) =

1x, �x� p(x, y , t) f (x) f (y)p. (dx) p.(dy) .

Again, we have the following upper bound for ltm U, ! ) : (3. 17)

ltm U , J ) $ ll /t/Jx, ll 2 11 /t/J x� lb exp

( �: t) -

·

We can rewrite the lower bound (3.16) in another way: (3. 1 8 )

hm (/, f )

>

k- !

e->.11/�fo + "[> - >. ; t Jf /;m - e ->.�t ll/t/Jx, ll 2

11/t/Jxm ll2

i= l

Now we can eliminate the middle term on the right-hand side of (3. 18 ) by choosing appropriate l and m. To this end, let us consider k+ 1 vectors fm = {![" , /2 , . . /;:_ 1 ) , m = 0, 1 , 2, . . . k in JRk - I and let us endow this {k - 1 )-dimensional space with a scalar product given by .

( v, w)

k- ! =

L v;w; e - >. , t . i=l

By using Corollary 3.9, out of any k + 1 vectors in (k - I)-dimensional Euclidean

space there are always two vectors with non-negative scalar product. So, we can find different l , m so that ( / 1 , f m ) � 0 and therefore we can eliminate the second term on the right-hand side ( 3 . 18 ) . Comparing {3. 17 ) and (3. 18), we have {3. 19)

e ->.• 1 ll /t/Jx, ll 2 11 / t/Jxm l l 2

$

f�fo

-

ll /t/Jx, ll 2 11 /t/Jxm lb ex

� - �:)

·

Similar to the case k = 2, we can choose t so that the right-hand side is at least � /J /0'. We select

t=

��

41

og 2

IJ/0'

11Nx, II 2 11Nxm 1! 2 •

54

3. DIAMETERS AND EIGENVALUES

From (3. 19), we have ,\

2 ll f t/Jx, llz ll f t/Jx� l l 2 < � k - t log !UO'

By substituting t from above and taking f

=

¢0 , (3.1 2 ) follows.

D

Although differential geometry and spectral graph theory share a great deal in common, there is no question that significant differences exist. Obviously, a graph is not "differentiable" and many geometrical techniques involving high-order derivatives could be very difficult , if not impossible, to utilize for graphs. There are substantial obstacles for deriving the discrete analogues of many of the known results in the continuous case. Nevertheless, there are many successful examples of developing the discrete parallels, and this process sometimes leads to improvement and strengthening of the original results from the continuous case. Furthermore, the discrete version often offers a different viewpoint which can provide additional insight to the fundamental nature of geometry. In particular, it is useful in focusing on essentials which are related to the global structure instead of the local conditions. There are basically two approaches in the interplay of spectral graph theory and spectral geometry. One approach, as we have seen in this section, is to share the concepts and methods while the proofs for the continuous and discrete, respectively, remain self-contained and independent . The second approach is to approximate the discrete cases by continuous ones. This method is usually coupled with appropriate assumptions and estimates. One example of this approach will be given in Chapter 10. For almost every known result in spectral geometry, a corresponding set of questions can be asked: Can the results be translated to graph theory? Is the discrete analogue true for graphs? Do the proof techniques still work for the discrete case? If not, how should the methods be modified? If the discrete analogue does not hold for general graphs, can it hold for some special classes of graphs? What are the characterizations of these graphs? Discrete invariants are somewhat different from the continuous ones . For ex­ ample, the number of vertices n is an important notion for a graph. Although it can be roughly identified as a quantity which goes to infinity in the continuous ana­ 2 and 2 n , for example. Therefore log, it is of interest to distinguish n, n log n , n , more careful analysis is often required. For Riemannian manifolds, the dimension of the manifold is usually given and can be regarded as a constant. This is however not true in general for graphs. The interaction between spectral graph theory and differential geometry opens up a whole range of interesting problems. • · ·

Notes This chapter is based on the original diameter-eigenvalue bounds given in (5 1] and a subsequent paper (53] . The generalizations to pairs of subsets for regular graphs were given by Kahale in (168). The generalizations to k subsets and to Riemannian manifolds can be found in (54, 55] .

CHAPTER 4

Paths, flows , and rout ing

4. 1 . Paths and sets of paths One of the main themes in graph theory concerns paths joining pairs of vertices. For example, the Hamiltonian path problem is to decide if a graph has a sim ple path containing every vertex of the graph. Some diameter and distance problems involve finding shortest paths. There are many basic problems depending on sets of paths that are either vertex-disjoint or edge-disjoint. These path problems arise naturally in a variety of guises, such as the study of communicating processes on networks, data flow on parallel computers, and the analysis of routing algorithms on VLSI chips. Some path problems appear to be quite difficult computationally. For example, the Hamiltonian path problem is well known to be NP-complete. The problem of finding disjoint paths between given pairs of vertices even in very special graphs (140) is also NP-complete. Nevertheless, we will see that eigenvalue techniques are amazingly effective in providing good solutions for a range of path problems. Before we proceed, we first define several types of disjoint paths that we c all flow, route set, and routing. C ons ider a graph G with vertex set V and edge set E. Suppose X and Y are two equinumerous subsets of vertices of G. In gener al , X and Y can be multisets and it is not necessary to require X n Y = 0 . For l X I = I Y I = m , a flow F from X to Y consists o f m paths i n G joining the vertices in X to the vertices in Y. We call X the input of the flow F and Y the output of F . Paths in F join vertices of X to vertices in Y in a one to- o ne fashion, but we do not care about "who is talking to whom." We do care that the paths be chosen so that no edge is overused. For example, t he paths might be required to be edge-disjoint or vertex-disjoint or with small con gestion in the sense that every edge (or vertex) of G is used in relatively few paths of F. We will define "congestion" precisely later. -

"

"

A route set is a flow with input-output assignments. Namely, for a specified assign ment A = { ( x ; , Y i ) : Xi E X, yi E Y} , a route set consists of paths Pi j oi n i n g x; to y; for each i . In other words, an assignment specifies "who is talking to whom." Roughly speaking, a routing R i s a dynamic version of a route set I t can b e defined as a pebble game. Initially, there is a p eb b le p ; placed a t each input vertex X ; with destination y; for each of the assignments (x; , y; ) i n A. At each time unit , a pebble can be moved to some adjacent vertex. The routing R is then a route .

55

56

4 . PATHS, FLOWS, AND ROUTING

set together with a strategy for moving pebbles to their destinations. Additional requirements can be imposed. For example, at each time unit, the edges used for moving pebbles should be (vertex- or edge- ) disjoint or all edges must have small congestion. Flow and routes are very useful in establishing lower bounds for Cheeger con­ stants as well as providing lower bounds for eigenvalues (see 4.2. and 4.5.) . Con­ versely, for graphs with good eigenvalue lower bounds, short routes and effective routing schemes exist with small congestion which will be described in 4.3. and 4.4. 4.2. Flows and Cheeger constants

Flows are closely related to cuts as evidenced by the max flow-min cut theorem which was used in the previous section. In fact, there is a direct connection between the Cheeger constants and flow problems on graphs. Although these observations are quite easy, we will state them here since they are useful for bounding eigenvalues. We follow the definition for Cheeger constants he and h(; as given in Section 2.2.

L EM M A 4 . 1 . For a graph G on n vertices, suppose there is a set of G) paths joining all pairs of vertices such that each edge of G is contained in at most m paths. Then

h (; = sup s

I E(S, S ) l > � min( I S I , l S I ) 2m -

PROOF. The proof follows from the simple fact that for any set S � V with l S I � l SI, we have I E(S, S) l · m

> >

0

As an immediate consequence, we have the following: COROLLARY 4 . 2 . For a k-regular graph G on n vertices, suppose there is a set P of G) paths joining all pairs of vertices such that each edge of G is contained in at most m paths in P. Then the Cheeger constant ha satisfies h e = inf s

�

I (S, S ) l

>

�

k mm ( I S I , l S I ) - 2mk

By using Cheeger's inequality in Chapter 2 and the above lower bound for the Cheeger constant derived from a flow, we can establish eigenvalue lower bounds for a regular graph. In fact, we can derive a better lower bound for At directly from a flow in a general graph. We first prove a simple version for a regular graph. THEOREM 4 . 3 . For a k-regular graph G on n vertices, suppose there is a set P of G) paths joining all pairs of vertices such that each path in P has length at most

4. 2 . FLOWS A N D CHEEGER CONSTA NTS l and each edge of G is contained in at most m paths in satisfies AI > -

P.

57

Then the eigenvalue A 1

n

kml

­

PROOF. Using the definition ( 1 .5) of the eigenvalues, we consider the harmonic eigenfunction f : V (G) --+ lR achieving ..X 1 • n

L { x ,y } E E (G)

(f( x) - f(y) ?

k L(f( x ) z,y

x, y

We note that for have

- f( y)) 2

E V(G) and the path

P ( x , y) joining x

d

y in G , we

e E P( z , y )

e E P ( x , y)

where f 2 ( e) = (f( x ) - /( y )) 2 for e = edges of G in P(x , y ) Hence

an

{x , y } , and JP (x , y ) J denotes the number of

.

L / 2 (e )

m

>

L L /2 (e ) x , y e E P(z,y)

eE E(G)

>

1

l L(f( x ) x ,y

- J ( y)) 2 .

Therefore we have n

kml ·

0

This completes the proof of Theorem 4.3 For a general graph, the above theorems can be generalized as follows:

THEOREM 4 . 4 . For an undirected graph G, replace each edge { u, v } by two directed edges (u, v) and (v, u) . Suppose there is a set P of 4e 2 paths such that for each (ordered) pair of directed edges there is a directed path joining them. In addition, assume that each directed edge of G is contained in at most m directed pa ths in P . Then the Cheeger constant he satisfies .

hc = mm PROOF. For any S

s:;:

JE(S, S) J vol G . > min(vol S, vol S) - 2m --

V(G) , we have

mJE(S, S ) J ;::: vol S vol S ;::: _

-

vol S voJ G 2

·

0

4. PATHS, FLOWS, AND ROUTING

58

THEOREM 4 . 5 . For an undirected graph G, replace each edge {u, v} by two directed edges (u, v ) and (v, u) . Suppose there is a set P of 4e 2 directed paths such that for each (ordered) pair of directed edges there is a directed path joining them, each of length at most l . In addition, assume that each directed edge of G is contained in at most m directed paths in P. Then the eigenvalue A1 satisfies vol G , ;:::: � -"1 The proof of Theorem 4 . 5 i s very similar t o that of Theorem 4.3 and will b e omitted. We remark that Theorems 4.3, 4.5 can be generalized in a number of ways. For example, instead of having one path joining two vertices, we can ask for a fixed number of paths or weighted paths with fixed total capacities ( in the spirit of the max flow-min cut theorem ) . Another direction is to derive the comparison theorems which will be discussed in Section 4.5.

4.3. Eigenvalues and routes with small congestion In a graph G, a random walk of length l starting at a vertex v of G is a randomly chosen sequence v = vo , Vt , . . . Vt , where each v;+l is chosen, uniformly at random and independently, among the neighbors of v; , for i = 0, . . . , l - 1. We say that the walk visits v; at time i.

10f1n

In a graph G with A1 > 0, a random walk starting from any vertex converges steps to the stationary distribution (if G is bipartite , we use a lazy roughly in random walk; see Section 1 . 5.) We will use thi s property to derive the following fact. THEOREM 4 . 6 . Let G be a graph on n vertices and suppose 1 2: log n / A 1 . Sup­ pose for any v E V (G) there are dv walks of length l starting at v . For any edge q, let I(q) denote the total number of walks containing q . Then, almost surely (i. e. , with probability tending t o 1 as n tends t o infinity), there is n o edge q s o that

I(q) > 101. PROOF. Let P denote the transition matrix defined by P(u , v ) =

{ 1/du0

if u an� v are adjacent otherwtse.

The probability that a random walk W ( u ) starting at u visits a vertex x at time i is precisely 1/Ju P i (1/Jx ) * where 1/Jy is the unit vector having 1 in coordinate y and 0 in every other coordinate. For a directed edge ( u , v ) , the probability that a random walk W (x) visits u at time i and v at time i + 1 is

4 . 3 . EIGENVALUES AND ROUTES WITH SMALL CONGESTION

59

With dv walks starting at v , the sum of the probabilities that there exists a walk W ( x) that visits q = {u,v} is 1-1

I(q )

L L d:xt/J:x Pi t/J: J du i=O z 1-1 L(lT) Pi t/J:/du i=O

<

where 1 denotes the all l's vector, T is the diagonal matrix with entries T(v, v ) = and

dv ,

(lT)Pi = lT. Therefore 1-1

< L lTt/J:/du i=O

l(q)

l.

Therefore, for each fixed v, the expectation of the random variable I( e ) is no more than l. We observe that this random variable is a sum of IE( G) I independent indicator random variables (see, e.g. [1 2] Theorem A . l 2 page 237) , and that for each fixed edge q , the probability that I(q) exceeds, say lOl, is at most e9 I 1

( 10 0 )

1

2

« n3

·

Since there are at most n edges, it follow that the probability that there is an edge with I(q) > lOl is much smaller than �- This completes the proof of Theorem 4.6. 0 The above estimate can in fact be proved directly. Suppose there is a set of m independent events such that the probability of the i-th event is p; . FUrthermore, suppose that L i � 'Y · Then the probability that at least s events occur is bounded by

P

L

S C { 1 , ... m } l SI =•

II i iESP

< �(LP;)" s.

< ee) s

•.

The proof of Theorem 4.6 follows by choosing 'Y

=

l and s =

10'""( .

TH EOREM 4 . 7. Let G denote a graph on n vertices. Let A = { (x ;, y;) : x; E X, y ; E Y} denote any assignment such that each vertex v is in X with multiplicity dv and in Y with multiplicity dv . Then there are paths P; joining X ; to y; of length at most }, log n such that each edge of G is contained in at most �� log n paths. PRO OF. Let P; denote a random walk of length 2l between x; and y; where l :::::: ���n . U sing an argument of Valiant in his work on parallel routing [243] (also see [40] ) , we may assume that each walk consists of two random walks of length

60

4. PATHS, FLOWS , AND ROUTING

l , one starting from x; and the other from y; . The reason for this is that by our eigenvalue condition, the distribution of the random walk of length l is close to its stationary distribution and hence one may view the walk P; as being chosen by first choosing its middle point (according to the stationary distribution) and then choosing its two halves. The proof of Theorem 4. 7 then follows from Theorem 4.6. 0 The above proofs for Theorem 4.6 and 4. 7 were adapted from the following simpler version for regular graphs in (7] :

T HEOREM 4 . 8 . Let G denote a k-regular graph on n vertices. Let 1r denote a permutation of the vertices of G Then there are paths Px joining x to 1r ( x ) of length 2 at most ;, log n such that each edge of G is contained in at most k �, log n paths.

4.4. Routing in graphs In this section, we consider a simple (though fundamental) problem of the following type: Suppose we are given a connected graph G with vertex set V and edge set E. Initially, each vertex v of G is occupied by a unique marker or "pebble" Pv . To each pebble Pv is associated a destination vertex 1r ( v ) E V, so that distinct pebbles have distinct destinations. Pebbles can be moved to different vertices of G according to the following basic procedure: At each step a disjoint collection of edges of G is selected and the pebbles at each edge's two endpoints are interchanged. Our goal is to move or "route" the pebbles to their respective destinations in a minimum number of steps. We will imagine the steps occurring at discrete times, and we let Pv (t ) E V denote the location of the pebble with initial position v at time t = 0, 1 , 2 , . . . . Thus, for any t , the set {Pv (t) : v E V } is just a permutation of V . We will denote our target permutation that takes v to 1r ( v) , v E V, by 1r. Define rt( G, 1r ) to be the minimum possible number of steps to achieve 1r. Finally, define r t ( G ) , the routing number of G, by

rt(G) = max rt(G, 1r) "

where 1r ranges over all destination permutations on G. (Sometimes we will also call 1r a routing assignment.) In more algebraic terms, the problem is simply to determine for G the largest number of terms T = ( Ut VJ ) ( u2v2 ) ( Ur Vr ) ever required to represent any per­ mutation in the symmetric group on n = l VI symbols, where each permutation T consists of a product of disjoint transpositions ( ukvk ) with all pairs { uk , vk } required to be edges of G. · ·

·

To see that rt( G) always exists, let us restrict our attention to some spanning subtree T of G. It is clear that if p has destination which is a leaf of T then we can first route p to its destination u, and then complete the routing on T \ { u} by induction.

4.4. ROUTING IN GRAPHS

61

An obvious lower bound on rt( G) is the following:

rt(G) � D(G) where D(G) denotes the diameter of G. For Pn a path on n vertices, our routing problem reduces to a well studied problem in parallel sorting networks, the so-called odd-even transposition sort (see [ 1 74] for a comprehensive survey) . In this case, it can be shown that rt(Pn ) = n. In fact, any permutation 1r on Pn can be sorted in n steps by labelling consecutive edges in Pn as e 1 , e2 , , en-l and only making interchanges with even edges en on even steps and odd edges e2k+I on odd steps. · · ·

Let Kn denote the complete graph on n vertices. In this case, because Kn is so highly connected, the routing number of Kn is as small as one could hope for (see [7] ) :

rt(Kn)

=

2.

For the complete bipartite graph Kn,n with n � 3, the following result is due to Wayne Goddard [ 143] .

rt(Kn,n ) = 4 . For any tree Tn on n vertices, it was proved in [7] that

rt(Tn) < 3n . However, the correct value of the constant may be half as large, as suggested by the following:

Conjecture. For any tree Tn on n vertices, rt (Tn ) :5 l

3(n - 1) 2

J.

Furthermore, equality holds only when Tn is a star Sn on n vertices. The following result on routing on the hypercube can be traced back to the early work of switching networks (see Benes [20] ) and has appeared frequently in the literature on parallel computing:

rt(Qn ) :5 2n - 1 . The exact value of rt(Qn) is still unknown. It is easy to see that rt(Qn) � n since the diameter of Qn is n. For small cases, it can be checked that rt(Qn ) � n + 1 for n = 2, 3.

Problem: Is it true that for the n-cube Q n , rt(Qn ) Perhaps, rt(Qn)

=

=

n + O(n) ?

n + o(n) for all sufficiently large n.

For the m by n grid graph Pm

rt(Pm

x x

Pn , m :5 n, Pn) :5 2m + n

.

4. PATHS, FLOWS, AND ROUTING

62

In

general, for the cartesian product of two graphs, we have [7, 1 78] r t( G

x

Note that since G x G' and G' the symmetric form

G') � 2rt(G) + rt( G ' ) .

x

G are isomorphic graphs, this can be written in

G') � min{2rt( G ) + rt ( G' ) , 2r t ( G' ) + rt( G ) . } Problem: Is it true that for every graph G, rt(G

x

rt(G

x

G) 2: r t( G ) ?

From the above results and partial results, we can see that the problem of de­ termining the routing number is quite difficult even for very special graphs. It is indeed surprising in a way that by using eigenvalues we can get very good approx­ imations for the routing number problem. The following arguments are basically adapted from [7] . In the remaining part of this section, we assume G is a regular graph.

T H EOREM 4 . 9 . Let G be a regular graph on n vertices and suppose l 2: log n/ A 1 . For each v E V, independently, let W ( v) denote a random walk of length l starting at v . Let I( v ) denote the total number of other walks W (u) such that there exists a vertex x and two indices 0 � i , j � 1, li - i l < 5, so that W(v) visits x at time i and W ( u) visits x at time j . Then, almost surely (i. e., with probability tending to 1 as n tends to infinity), there is no vertex v so that I(v) > lOOl .

The proof is very similar to that of Theorem 4.6 and is omitted. T H EOREM 4 . 10. Let G denote a regular graph on n vertices and let a be a permutation of order two on V (i. e., a product of pairwise disjoint transpositions) . Put l = !? log n . Then there is a set of n walks W(v), v E V, each of length 21, where both W (v) and W (a(v)) connect v and a(v) and traverse the same set of edges {in different directions) satisfying the following: If l(v) denotes the total number of other walks W (u) such that there exists a vertex x and two indices 0 � i , j � l, li - il < 5, so that W(v) visits x at time i and W(u) visits x at time j or at time 21 - j, then I(v) � 4001 for all v . T H EOREM

4 .11 . Let a denote a permutation o n the vertex set of G. Then

r t (G, a ) = 0 (

1

A1

l og2 n) .

P ROOF . Let G denote a regular graph on n ve r tic es . It suffices to consider a permutation u of order two on V since any permutation is a product of at most two such permutations (as proved in the proof of r t ( Kn ) = 2; also see [7] ) . We set l = l? log n . We want to show that r t( G, u ) = 0(12 ) . Let W(v) be a system of walks of length 21 satisfying the assumptions of the previous theorem. Let H be the graph whose vertices are the walks W ( v) in which W ( u ) and W ( v) are adjacent if there exists a vertex x and two indices 0 � i , j � 1 , li - i l < 5 so that W (v) visits x at time i and W (u) visits x at time j or at time 21 - j . Then the maximum degree of H is 0(1) and hence it is 0(1)-colorable. It follows that one can split all our p at h s W ( v ) into 0( l) classes of paths such that the paths in each class are not adjacent in H. Consider now the following routing algorithm. For each set of paths

4 . 4 . ROUTING IN GRAPHS

63

as above, perform 2l steps, where the steps numbered i and 2l + 1 - i correspond to flipping the pebbles along edges numbered i and 2l + 1 - i in each of the paths in the set for all 1 � i � l. One can check that by the end of these 2l steps, the ends of each path exchange pebbles, and all the other pebbles stay in their original places. (Note that some pebbles that are not at the ends of any of the paths may move several times during these steps, but the symmetric way these are performed guarantees that such pebbles will return to their original places at the completion of the 2l steps. ) By repeating the above procedure for all the path-classes, the 0 result follows.

We mention here several problems closely related to the routing number of a graph. One such problem is the following: Suppose G = (V, E ) is a connected graph on n vertices. For a permutation 1r , we consider a route set P, which is just some set of paths Pi joining each vertex vi to its destination vertex 1r ( vi ) , for i = 1 , · · · , n. For each edge e of G, we consider the number rc(e, G, 1r , P) of paths Pi in P which contain e. The route covering number rc( G) of G is defined to be

rc(G) = max min max rc(e, G, 1r, P) . " P

eE E

I n other words, for each permutation we want t o choose the route set s o that the maximum number of occurrences of any edge in the paths of the route set is minimized. For example, for the n-cube Qn , the method (7] used to establish the route set gives

rc(Qn) � 4. In the other direction, by choosing 1r to be the permutation of vertices in Qn so that the distance between v and 1r ( v ) is n for every vertex v, it can be easily seen that

Conjecture: rc(Qn )

=

2.

Also of interest is a "symmetric" version of the route covering problem, espe­ cially for Q n : A pairing for Qn is a partition of the vertex set of Qn into subsets of size 2 . Is it possible to find edge-disjoint paths joining vertices of each pair for any pairing of Qn ? The answer is negative when n is even. However, for odd n t h is problem remains open .

4 . PATHS, FLOWS, AND ROUTING

64

4.5. Comparison theorems We can often bound the eigenvalues of one graph by the eigenvalues of another provided pairs of adjacent vertices in the first graph can be joined by "short" paths in the second graph. Although the proofs for these comparison theorems are quite easy, the applications are abundant. Interesting examples along this line are given in Diaconis and Stroock (98] and numerous other papers (99, 100, 125] for comparing various different card shuffling schemes. We remark that the comparison theorems in this section can be viewed as generalized versions of the so-called "Poincare" inequalities (98] . THEOREM 4 . 1 2 . Let G and G' be two connected regular graphs, with eigenval­ ues At and .X� and degrees k and k' , respectively. Suppose that the vertex set of G is the same as the vertex set of G' . We assume that for each edge {x, y } in G, there is a path P(x, y) in G' joining x and y of length at most l. Furthermore, suppose that every edge in G' is contained in at most m paths P( x, y) . Then we have I

\ > At -

kAt k ' lm "

-­

P ROOF. Using the definition of the eigenvalues, we consider the harmonic eigen­ function f achieving At in G' .

1:

.X'l

(J(x) - / (y))2

{ z , y } E E(G') =

k' L f 2 (x) k

1:

(j(x) - /(y) ) 2

L

( f (x) - /(y)) 2

{ z , y } E E(G' )

=

k'

L

(J (x) - f( y) ?

{ z ,y } E E(G)

k

{ z ,y } E E(G)

L /2 (x)

We note that for {x, y} E E(G) and path P (x, y) joining x and y in G' , we have

(f(x) - / (y)) 2 � I P( x , y ) l

L

/ 2 (e) � l

e E P(z , y)

L

P (e)

e E P( z ,y)

where /2 (e) = ( f (x) - / (y)) 2 for e = {x, y} , and I P (x, y)l denotes the number of edges of G' in P(x, y ) . Hence m

L:

/ 2 (e)

>

e E E(G')

L: L P (e) L (J (x) - f(y)) 2 .

{ z ,y} E E(G) e E P( z ,y) >

1 l

{ z , y } E E(G)

4 . 5 . COMPARISON THEOREMS

65

Therefore we have { z , y } E E(G)

k' lm

>

(f(x ) - f(y ))2

L

k

k A k ' lm J .

L J 2( x ) k

This completes the proof of Theorem 4. 12.

D

It is not surprising that the above proof is quite similar to some of those in Section 4.2. There are several generalizations of Theorem 4.12: THEOREM 4 . 1 3 . Let G and G' be two connected gmphs, with eigenvalues A1 and A�, respectively. Suppose that the vertex set of G is the same as the vertex set of G' . Assume that for each edge { x, y } in G, there is a path P ( x, y ) in G' of length at most l , and for each vertex v, the degree dv of v in G is at least ad� , where d� is the degree of v in G' . Jilurthermore, suppose every edge in G' is contained in at most m paths P (x, y ) . Then we have

, , aA 1 .��. 1 >- lm ·

Instead of proving Theorem 4. 13, we will prove the following generalization: THEOREM 4 . 14. Let G and G' be two connected gmphs, with eigenvalues A 1 and A�, respectively. Suppose that the vertex set of G can be embedded into the vertex set of G' under the mapping r.p : V (G) --t V (G' ) . Suppose r.p satisfies the following conditions for fixed positive values a, l, m : ( a ) : Each edge {x, y} in E(G) is associated with a path, denoted by P., ,11 , join­ ing r.p ( x ) to cp(y) in G ' of length at most l . (b) : Let dv, d� denote the degrees of v in G and in G' , respectively. For any v in V (G'} , we have

L d., 2:: ad�. z e , - • (v)

(c) : Each edge in G' is contained in at most m paths P., ,11 • Then we have

PROOF. The proof is very similar to that of Theorem 4.12. For a harmonic eigenfunction g of G' , we define f : V (G) --t IR as follows: For a vertex x in V (G) ,

f( x ) = g( r.p (x) )

-

where the constant c is chosen to satisfy

L f(x)d.,

=

0.

c

4. PATHS, FLOWS, AND ROUTING

66

We note that

L / 2 ( x ) dx

L (g(

=

x E V (G)

x E V(G)

L (g( v) - c)2 L dx

=

v E V(G' )

(4 . 1)

>

a

rp - l (v)=x

L g(v)2 d� .

vEV(G')

Now, for {x, y} E E(G) with

Px , 11

denote the path corre­

(g(u) - g(v))2 :5 I Px ,11 1 L g2 (e ) :5 l L g2( e ) , e E Pz .v

eEPz . v

where g2 ( e) =

(g(a ) - g( b))2 for e = {a, b}.

Hence we have

L g2 ( e ) � L mg2(e )

m

e E E(G' )

e E E( G ' )

:L

:L l( e )

L

L y(g(u) - g(v) )2

{ x , y } E E( G) e E P( x ,y)

>

1

{ x ,y} E E(G) u =rp (x) v =rp ( y)

(4.2)

�

�

L {z ,y } EE( G)

(f( x ) - f(y)} 2 .

Combining inequalities (4.1),(4.2) , we have

L , A ,1

=

{ u, v } E E(G' )

(g(u ) - g(v))2

sup ����-------t L (g(v) - t ? d� vEV(G')

L �

{ u,v } E E(G' )

(g(u) - g(v) )2

L (g( v ) - c)2 d�

v E V (G' )

L =

{ u , v } E E ( G' )

L { z , y } E E( G )

>

1

ml

(g( u ) - g (v )?

L { z ,y } E E(G)

(f(x ) - f(y )?

(f(x ) - f(y))2

L (g(v) - c) 2 d�

v E V (G' )

L { z , y } E E( G )

(f( x ) - f(y)?

L J 2 (x ) d,

x E V(G)

a

4.5. COMPARISON THEOREMS

67

by using

L

Z E
/ 2 (x) dz � (g(v) - c) 2 d� .

Since z we have

L

{z,y}EE(G)

(f(x) - f (y ) ) 2

---==------

..:...._

L

z EV(G)

/ 2 (x ) dz

� AI ·

Hence ,

a

A l � ml A I

and the proof of Theorem 4.14 is complete.

0

We remark that Theorems 4.3 and 4.5 in Section 4.2 are just special cases of Theorem 4.14 in which G is taken to be a complete graph and G' is chosen arbitrarily. We also remark that the generalized version in Theorem 4.14 can often give stronger results for certain problems. For example, we consider the following simple and natural random walk problem on generating sets of groups which arises in computational group theory.

EXAMPLE 4. 1 5 . Let H denote a graph on n vertices each of which is labelled by an element of a group r. At each unit of time, one of the vertices, say v with label g, can be changed to g f where f or f - 1 is a label of a neighbor u of v. Suppose we start with the case that the set of all vertex labels generates the group r. The problem of interest is to determine how rapidly this processes mixes , i.e. , how many steps it requires to be close to a "random" generating set.

By using Theorem 4. 14, we can obtain an upper bound of the form cDn 2 , where c depends only on the size of r, and D denotes the diameter of H (see [80] ) . Similar bounds have also been obtained by Diaconis and Saloff-Coste in [102] using more complex comparison techniques. However, all of these bounds are rather far from what is believed to be the truth, namely, that order n log n steps (under total variation distance) . In [8 1] , it is proved that in fact this bound is achieved for the case that H = Z 2 . Interestingly, for relatively pointwise distance, O(n 2 ) is proved to be the correct bound [81].

Not es Path arguments has been used early on to compute isoperimetric constants of a graph. For example, in the early work of Bhatt and Leighton [33] on VLSI design and parallel computation, path arguments were extensively lltilized. Jerrum

68

4. PATHS, FLOWS, AND ROUTING

and Sinclair [166] used path arguments to bound the Cheeger constant in order to bound the eigenvalues in their seminal work of estimating permanents. Diaconis and Stroock (98] used path arguments to directly bound eigenvalues. Sections 4.3 and 4.4 on paths and routing are mainly based on [7] . Some variations of the comparison theorems in Sections 4.2 and 4.5 can be found in [80] .

CHAPTER 5

Eigenvalues and quasi-randomness

5 . 1 . Quasi-randomness The isoperimetric properties and eigenvalue inequalities in previous sections are quite effective in dealing with many aspects of graphs with At bounded away from 0 , i.e. , At � c > 0 for some absolute constant c. B asically such graphs have "small" diameters and "large" vertex and edge expansions. Furthermore, there are "many" paths with "small" congestion joining given pairs of vertices as well as various dynamic routing schemes for sending messages (or pebbles ) along the paths simultaneously. The main questions concern how "small" or "large" these quantities can be. Many answers lie in estimating these quantities in terms of eigenvalues ,

.

When all A; are close to I , the graphs satisfy additional nice properties which will be discussed in this chapter. For example, a random graph has eigenvalues I I A; I = 0( j;; ) for i =f:. 0. (Here we can use any model of a random graph (30] ) . In other words almost all graphs satisfy 1 1 - .X; I 0( -jn ) for all i =f:. 0. For sparse graphs with average degree k, the corresponding condition is I I - .X; I = 0( � ) . -

=

,

There is a long history in extremal graph theory for investigating how one graph property relates to another In recent years, there has been a great deal of development in the study of random graphs which analyzes the behavior of almost all graphs These two areas which might seem to be quite different , have a great deal of overlap in the following sense: Most graph properties which are satisfied by a random graph are related in a strong way. There is a series of papers [67] (70] [71] [72] [ 73] [74] [75] [76] (224] introducing a large equivalence class of graph properties, in the sense that any graph which satisfies any one of the properties must satisfy all of them. Graphs that satisfy these properties are said to be quasi­ random. One of the main quasi random properties concerns the concentration of all nontrivial eigenvalues. So, such an eigenvalue distribution dictates much of the behavior of the graph. .

,

.

-

As an example, here we illustrate the following list of quasi random properties for a graph G with edge density I/2 [76] : -

P1 :

max 1 1 i ;tO

-

A; I

=

o( 1) and G is almost regular ( i.e . , all except o ( n) ver tices

have degree n/2 + o ( n )) . P2 :

For each subset S � V (G) , the number e(S) of edges of G with both end69

5. E I GENVALUES AND QU ASI-RANDOMNESS

70

point in S satisfies e (S)

P3 :

:

l 1 SI2 + o(n2 ) .

For each subset S � V (G) with lSI = L¥-J , e (S)

P4

=

For

u

=

U6 + o( 1 ) )n2 •

in V (G) , let

N ( u ) denote the neighborhood {

v : u

"'

v

} of u .

'E I N( u) n N(v) - � I = o(n2 ) . u,v

P5(s) :

For s � 4 and for all graphs M(s) on s vertices, the number N0(M(s)) of "labelled" induced subgraphs of G isomorphic to M(s) satisfies

N(;(M(s)) = ( 1 + o( 1))n"T ( ;) . P6 (t) : For the 2t-cycle C2 1 , t � 2 , the number Na (C t ) of occurrences of C t 2 2 as a (labelled) subgraph of G is

Na ( Cu ) = ( 1 + o( 1 ) ) ( ¥" ) t , and

e (G) � (1 + o( 1 ) ) � . 2

We remark that the description of our graph properties typically contain the o( ·) notation. The statement P(o( 1 ) ) => P' (o( l ) ) means that for any t > 0, there exists 8 such that P(8) => P'(f) . Two properties P and P' are equivalent if P => P' and P' => P. In [76] , it was shown that all Pi , i = 1 · · · 6, are equivalent. The list of quasi-random properties is still increasing (see [224] ) and each new addition further strengthens the strong consequences of the equivalence. Although some properties are easy to compute (such as P1 , P4 , P6 ) , some are (at present) computationally intractable (e.g. P2 , P3 , P5 ) . We can construct graphs satisfying all the properties by verifying only one property. Thus, this provides a validation scheme for approximating one difficult property by using another equivalent prop­ erty which is easier to compute. It is also desirable to have a quantitative estimate for the error in the o( ·) term , which is often the focus of many extremal graph problems. The main goal of this chapter is to give a general unified treatment of quasi­ random graphs. Throughout , the eigenvalue property will be central and the goal will be to bound the o( 1 ) estimates in each of the properties in terms of .X = max 1 1 Ai I , if possible. We will examine several major properties with emphasis

i ,tO

-

on their relations to the eigenvalues.

5 . 2 . THE DISCREPANCY PROPERTY

71

5.2. The discrepancy property Let G denote a graph having a vertex set V with n vertices and an edge set E with e edges. The edge density p is defined to be 2e/n 2 . For subsets X, Y c V , we recall the notation E(X, Y) as the set of ordered pairs corresponding to edges with one endpoint in X and the other in Y, i.e. , E(X, Y )

=

{ ( u, v ) : u

E

X, v E Y and {u, v} E E}.

Here, X and Y are not necessarily disjoint. We denote e(X, Y) =I E(X, Y) I .

For a subset S of V, the discrepancy of S, denoted by disc(G, S) , is defined to be disc(G, S) = le(S, S) - piSI 2 1

The a-discrepancy of G is the maximum discrepancy of S � V over all S with I S I = a n , where 0 < a :::=; 1 , i. e . , disc(G; a ) =

max disc(G, S) . ISI= Lo n j In particular, the discrepancy of a graph G, denoted by disc G is just disc G

=

max disc(G; a ) . 0

In a certain sense, the discrepancy is the "quantitative" version of the Ramsey property which asserts that when a is very small ,...., c ':s the a-discrepancy can 2 be as large as piSI . In general, the problem of determining the a-discrepancy is a very difficult problem and is known to be NP-complete. It is therefore of interest to derive upper bounds for the discrepancy using other methods, e.g. , by eigenvalue arguments.

(

'1/J s , defined by

n) ,

For a subset S of the vertex set V of G, we consider the characteristic vector

'1/Js( u)

=

We note that

{0 1

if u E S, otherwise.

L L Au v = e (S, S)

('1/Js , A'I/Js } =

uES vES where A is the adj acency matrix of G.

Also, the edge density satisfies p=

( 1 , A 1} ( 1 , J1}

and l S I = ( '1/Js , 1 } where J denotes the all

1 's matrix.

If 1 is an eigenvector of A (as is the case for regular graphs) , we can then suc­ cessfully bound the discrepancy by the bounds on the eigenvalues of the adjacency matrix. However, for general graphs, 1 is usually not an eigen vector for A. These

72

5. EIG ENVALUES AND QUASI-RANDOMNESS

obstacles can be overcome by considering the eigenvalues of the Laplacian following:

THEOREM Then

5. 1 .

Suppose X, Y are two

je(X, Y) where X

=

vol

max; ;eo 1 1 - A; I ·

��;I =

< X v'vol X vol Y

yI

1/Jx T 1 1 2 ( I - C) T 1 1 2 1/Jy .

Suppose T l / 2 1/Jx =

L a;r/J;,

Tl/2 1/Jy =

L b;r/J;,

i

i

where rjJ; 's are eigenvectors of C and, in particular, ¢Jo Since r/J; , i � 1 , is orthogonal to 4>0 , we have JT 1 f 2 ¢J; = 0

=

T 1 12 1 / v'vol G.

for all i � 1 . Also,

Therefore, we have

disc( X, Y)

- ao bol A 1/!y - ao bo l

=

je(X, Y)

=

11/Jx

=

11/Jx Tl / 2 (I - C -

=

I L a; b; (l i� l

< <

X

- A; ) l

T l f 2 JT 1 f 2 ) T l /2 1/J•Y I vol G

L a ; L b;

i� l i� l v'vol X vol X vol Y vol Y X vol G

since

L a� = ao -

_

(A..'I'O , T 1 1 2

in the

su bsets of the vertex set V of a graph G.

PROOF. Following the above notation, we have e(X, Y)

as

·'·

vol X

'I'X

)_ vol X v'

vol G

""a2 L....t

i�l

•

=

5. 2 .

THE DISCREPANCY PROPERTY

vol G -

(vol X)2 vol G

vol X vol X � vol X vol G

73

·

This completes the proof of this theorem.

D

In fact, we have shown a slightly stronger result:

THEOREM 5 . 2 . Suppose X, Y are two subsets of the vertex set V of a graph G . Then ). v'vol X vo lY vol l X vol y < (5. l ) l e ( X Y) I l 0 5. where = maxi # I I >.i I · ,

��

_

� ��

-

Here we state a few consequences of Theorem 5 . 1 :

COROLLARY 5 . 3 . Suppose X is a subset of vertices i n a graph G . Then ). vol v lX X l e (X, X ) � � vol X . l COROLLARY 5 .4 . Suppose X is a subset of vertices in a k-regular graph G . Then -

(v:�ld2 1

le ( X , X )

��

- ki X I 2 1 � k5.1 X I . n

We note that Theorem 5.2 is closely related to the edge-expansion properties that we discussed in Chapter 2. If we can find an upper bound for the number of edges both ends of which lie inside a set X, then we can find a lower bound for the number of edges leaving X . Using Theorem 5.2 and Corollary 5.3 , we have the following isoperimetric inequality.

COROLLARY 5 . 5 . Suppose X is a subset of vertices in a graph G. Then the edge boundary ax satisfies I 8X I > {l 5. ) vol x vol G . vol X _

PROOF. We consider

vol X = e ( X, X) + I8X I .

5. ) vol x

By substituting using Corollary 5.3, we have

I8X I > {l vol X -

_

vol G .

D

From the statement of (5. 1 ) , it is tempting to make a number of conjectures. Some of these questions can be partially answered, but most of them are unresolved.

Qu estion 1: Suppose a graph G satisfies, for some fixed a , v'vol X vol Y vol X vol Y vol X vol Y le{X, Y) I�a vol G G vol

5. EIG ENVALUES AND QUASI-RANDOMNESS

74

for all X, Y � V (G) . Is it then true that A � 100a? (Of course, 1 00 can be replaced by 10100 , if this is easier.) Suppose we use Corollary 5.5 so that we get h a > 1 2" . Using Cheeger's inequality from Chapter 2, we have

h'i; ..:.__ ( 1 - a)2 ....:._ > 8 - 2

,\ 1 >

-

-

We therefore have shown that 1

-

>. 1

< -

1

-

( 1 - a? 8

This is still quite distant from the desired bound A � lOOa. However, it is evidence in support of an affirmative answer to the above question.

Question 2: Suppose a graph G satisfies, for some fixed a,

( vol X vol Y v'vol X vol Y vol X vol Y I �a I e X, y ) vol G vol G

(5-2)

for all X , Y � V (G) . Is it true that

disc G �

100a ?

In other words, for all X, Y

� V(G) ,

does the following equality hold:

l e(X , Y ) - pi X I I Y I I � 100a

where

p

=

v�2°

�� �ol

v'vol X vo

is the edge density of G?

X vol

y

This question can be answered in the negative. We now construct a graph satisfying the above assumption but having a large discrepancy. In fact, the graph is not even "almost regular."

Let H be a graph with vertex set A U B where I A I = � , IBI = � and A n B = 0. The induced subgraph on A will be a random graph with edge density 1 / 2 . The induced subgraph on B will be a random graph with edge density 1/4. The bipartite subgraph between A and B will have edge density 1/3. It is left as an exercise to check that H satisfies (5.2) , but the discrepancy is large (about cn 2 where c 2: 0.01 ) .

It i s of course true that i f a graph is "almost regular" and i t satisfies (5.2) , then its discrepancy is small. "Almost regular" is a necessary condition for quasi­ randomness. In the fourth section of this chapter, we will examine how the discrep­ ancy relates to other quasi-random properties. Theorem 5.2 gives a good approximation for the discrepancy of many families of graphs. For example, many expander graphs (see Chapter 6) and the random graphs (as described in Remark 1) have eigenvalues 1

>. -

Vk

5. 2 .

THE DISCREPANCY PROPERTY

75

where k is the average degree. For such graphs G, the above theorem implies disc(G; o:)

:=

for some absolute constant c.

sup le(X, X) - p iX I2 1 � cv'ko:n

I X I = <>n

In the book of Erdos-Spencer [120] , a lower bound for the discrepancy of any graph G with edge density 1/2 was given by disc G � cn31 2

for some absolute constant c. So, the eigenvalue upper bound for the discrepancy is within a constant factor of the best possible value for many graphs. The discrepancy of a random graph can be easily estimated. Here is a sketch of a proof that disc(G; a) "' cn31 2

for a random graph G on n vertices where c is a constant depending only on the edge density. Here we assume that the edge density p is a fixed positive quantity when n approaches infinity. Let G denote a random graph with edge density p. We define a function f which assigns the value ( 1 - p) to every edge of G and the value - p to every non­ edge of G. It is easy to see that I L u , vE S f (u, v ) I = di sc(G, S ) . Using the Chernoff bound (see (12] , pp. 237} , the probability that the random graph has discrepancy more than (3 satisfies Prob(di sc(G; a} > (3) � exp( -(32 /(2 pa2n2 ) ) .

Therefore the total probability of having some set of size o:n for which the discrep­ ancy is (3 is at most (: ) exp( -f32 / (2po:2 n 2 )) . n

When the above quantity is smaller than 1 , there must exist a graph with discrep­ an.cy no more than (3. Indeed, we can choose (3 to be c o:n312 for some appropriate constant c so this is true. The exact expression of c in terms of o: and p are not hard to derive. (Hint, try c = a JpH (a) where H (x) is the "entropy" function.) How will the discrepancy disc(G; a} behave when a is small, say, for example, a n is smaller than y'n for a graph with edge density 1/2? This , in fact , embodies a wide collection of classical combinatorial problems for the whole range of subset sizes an, say, from 0 to n , and/or for any edge density. For most of such problems, our knowl edge is quite limited and the known tools are few. Perhaps the only powerful method is to use eigenvalues to upper bound the discrepancy for regular graphs when a > >. as demonstrated in Corollary 5 .4. For general a, such as a < >., the discrep ancy disc(n; a}

is not well understood.

=

sup

! V {G) I = n

disc(G; a )

5. EIGENVALUES AND QUASI-RANDOMNESS

76

It would be tempting to define discrepancy s �p

as

I e(X, X) - piX I 2 1 IXI2

However, from classical Ramsey theory we know that any graph on n vertices contains an induced subgraph on a subset X of c log n vertices which is either a complete subgraph or an independent set. Thus, l e (X, X) - piX I2 1 could be as large as pj X I 2 . In a way, discrepancy problems can be viewed as a qualitative general­ ization of Ramsey theory. The discrepancy is concerned with induced subgraphs of all sizes while Ramsey theory focuses on the containment of special subgraphs (which can be small subgraphs with large discrepancies ) . Most bounds like the ones above are proved by using probabilistic methods. Explicit constructions are quite poor in their performance in comparison with the random graphs. The reader is referred to several excellent papers and surveys by Thomason (234, 235, 236] on this subject under the name of " (p, a ) - jum bled'' graphs. For the lower range of a , the discrepancy problems are basic Ramsey problems which we will discuss briefly in the next subsection.

5 . 2 . 1 . The Ramsey property. A fundamental result of Ramsey (21 5] guar­ antees the existence of a number R(k, f) so that any graph on n � R (k, f) vertices contains either a complete graph of size k or an independent set of size e. The problem of determining R(k, f) is notoriously difficult. The first non-trivial lower bound for R(k, k ) , due to Erdos (1 1 1] in 1947, states (5.3)

R ( k , k ) > ( 1 + o(l)) 1"" k ev

2 2 "12. ·

In other words, there exist graphs on n vertices which contain no cliques or inde­ pendent sets of size 2 log n when n is sufficiently large. The proof for (5.3) is simple and elegant , and is based on the observation that the probability of having a clique or independent set of size k is at most

(�) . 21 - ( � ) . We see that if this quantity

is less than one, there must exist a graph without any clique or independent set of size k . This basic result plays an essential role in laying the foundations for both Ram­ sey theory and probabilistic methods, two of the major thriving areas in combina­ torics. In the 40 years since its proof, the bound in (5.3) has only been improved by a factor of 2, again by probabilistic arguments [229] . Attempts have been made over the years to construct good graphs (i.e., with small cliques and independent sets) without much success (74, 144] . H.L Abbott (1] gave a recursive construction with cliques and independence sets of size cn1 o g 2/ log 5 . Nagy (201] gave a construction reducing the size to A breakthrough finally occurred several years ago with the result of Frankl (128] , who gave the first Ramsey construction with cliques and independent sets of size smaller than for any k. This was further improved to ec(log n)314 (log log n) • l • in (64] . Here we will outline a construction of Frankl and Wilson (1 30] for Ramsey graphs with cliques and independent sets of size at most ec(log n log log n ) 1 1 2 •

cn113.

n 1 /A:

5.3.

THE DEVIATION OF A GRAPH

77

EXAMPLE 5.6. Let q be a prime power. The graph G will have vertex set { F � { 1 , , m } : I F I = q2 - 1 } and edge set E = { ( F, F' ) : I F n F' I� 1 - ( mod q) } . A result i n [130) implies that G contains n o clique o r independent set 3 of size q '.:' . By choosing m = q , we obtain a graph on n = q2 � 1 vertices V

=

· ·

·

( 1)

containing no clique or independent set of size e c (log n

log

(

log n) ' 1 2 .

)

The proof involved in the above construction is based on a beautiful result of Ray-Chaudhuri and Wilson [21 7] on intersection theorems. This type of inter­ section graph and the related intersection theorems provide excellent examples for many extremal problems including the discrepancy problems. A graph which has often been suggested as a natural candidate for a Ramsey graph is the Paley graph (see more discussion in Section 6) . Very little is known about its maximum size of cliques and independent sets. For the lower bound, a result of S . Graham and C. Ringrose [145] shows that infinitely many Paley graphs on p vertices contain a clique of size c log p log log log p. (This contrasts with the trivial upper bound of c.;p.) Earlier results of Montgomery [1 99] show that assuming the Generalized Riemann Hypothesis, we would have a lower bound c log p log log p infinitely often. If we take the Ramsey property as a measure of "randomness," the above results show that the Paley graphs deviate from random graphs. There is no question that the problem in constructive methods for which a solution is most widely sought is the following, posed long ago (as early as the 40's) by Erdos: Problem : Construct graphs on set of size c log n .

n

vertices containing no clique and no independent

Instead of focusing on the occurrence of cliques and independence sets, similar problems can be considered on the occurrence or the frequency of other specified subgraphs [35 , 146, 2 19, 249) . It is not difficult to show that almost all graphs contain every graph with at most 2 log n vertices as an induced subgraph. The best current constructions containing every graph with up to c y'IOgTi vertices as induced subgraphs can be found in [73, 1 29] .

5.3. The deviation of a graph We h ave discussed various aspects of the discrepancy of a graph. In spite of the important role discrepancy plays in various extremal problems, one major chal lenge is that discrepancy is difficult to compute since its definition involves taking the extremum over all choices of subsets with potentially exponentially many cases. Here we will consider another invariant, the so-called deviation of a graph. Alt hough its definition seems to be more complicated, it can be easily computed in p oly nomial time. Furthermore, deviation is very closely related to discrepancy and it can be used to prove upper and lower bounds for discrepancy.

5 . EIGENVALUES AND Q U ASI-RANDOMNESS

78

x

:

For a graph G with edge density V x V --+ JR. as follows:

x( x , y ) = For a 4-cycle C, we denote

we define a weighted indicator function

p,

{ 1 --pp

x(C)

if x '"" y otherwise.

II x( x , y )

=

{ z ,y } E C

and we define the deviation of G by

dev G

=

1 p4 n4

� x(C)

"' 1 4 n4 L; x ( x , y ) x ( y , z )x (z , w)x ( w, x) x,y,z,w

p

where x , y, z, w range independently over all vertices of G. For the special case of p = 1 /2, the deviation is exactly 1 / 16n 4 times the number of "even" 4-cycles minus the number of "odd" 4-cycles. (A 4-cycle { x , y, z , w} is said to be "even" if x ( x , y ) x ( y, z)x ( z, w) x( w, x) is positive. ) Before we derive relations between deviation and eigenvalues, we want to give a quantitative measure of regularity and irregularity in a graph. ( Recall that "almost regular" means that all but o(n) vertices have degree within ( 1 + o( 1 ) ) of the average degree of the graph ) . We define the irregularity of a graph G, denoted by irr G as follows: irr G := !!..

n

where j V (G) I

=

n, Pv

=

L ( _!__ - .! ) v

v E V (G)

P

P

dv /n, and p = � dv /n 2 .

The smaller the value of irr G is, the more closely the graph G approximates a regular graph. When irr G = 0, G is regular, as shown by the following useful lemma. The discussions and techniques in this section can be greatly simplified if we only consider regular graphs. However, in the same spirit of preceding sections, we consider a general graph for completeness. LEMMA 5 . 7 .

For any A � V(G) ,

"' ( _!__ _! ) L;

v E V (G)

_

Pv

p

>

-

we have

( vol A - 1Aipn) 2 p2 n vol

A

for any A � V (G) .

5 . 3 . THE DEVIAT I O N OF A GRAPH

79

P ROOF. 1 - 1) L (p v P -

vE V( G )

For P A

=

L Pv / I A I , we have v

- -) � o. L(Pv PA 1

1

v EA Therefore,

1 - -1 ) L( Pv

L( _!_ Pv

p

vE A

vE A >

_

_!_ + _!_ - � ) PA

2 ) 2_ - p�)

PA

P

vE A PA 1 1 IAI · ( - - ) PA P in � ) A l IAI ( vol A p ( I Aipn - vol A ) IAI. p vol A -

_

Therefore we have

L

( __!__

vE V( G ) Pv

_

� > (vol A - I A i pn) 2 ) . p2 n vol A p -

The lemma is proved.

0

The above lemma implies the following useful facts:

COROLLARY 5 . 8 . I vol A - I A ipn I :S Virr G vol G

for all A � V (G) .

LEMMA 5 . 9 .

5 . EIGENVALUES AND QUASI-RANDOMNESS

80

P ROOF . From the proof of Lemma 5.7, we have

1 1 "' L.)- - -) vEA Pv p

"' ( _!_ ! ) + I Aipn - vol A L-p2 n vEV(G) Pv p . G + Jpn2 vol A irr G n-zrr p2 n P

�

_

<

'!!:. irr G + '!!:. Jirr G.

p

p

Therefore, Lemma 5.9 is proved.

0

The deviation dev G can be expressed in the following form.

THEOREM 5 . 10 .

dev G =

where

Nz = N( x ) = {y : y x } �

� L(INz Pn , zy

and dz

=

n

N111 - p2n ) 2

INz l ·

L x( x , z )x( y , z)x ( x , w )x(z, w ) z,y

z

z,y +IN, n N11 I P2 ] For fixed =

x, y ,

we have

INz n Ny i ( I - p) 2 - INz n NyiP(1 - p) - INz n Nylp( 1 - p) + INz n Ny i P2 (IN. n Ny l - p2 n ? - ( I Nz l - pn)p - ( INy l - pn) p.

Therefore we have

and Theorem

Z , JI

5 . 10 is proved.

0

The above Theorem is useful for relating the neighborhood property ( e.g. , P4 ) to the deviation property.

THEOREM 5. 1 1 .

For a regular graph G, we have dev G =

L(l - A; ) 4 i#O

PROOF. We consider the matrix

1 _ Tlf2J T lf2 • M :.=. I - C - _ ( ) vol G

5.4. QUASI-RANDOM G RAPHS

whose eigenvalues are exactly 0 and 1 trace of M4 , we obtain

-

A; for i

=

1, . . . , n

81

-

1. Considering the

(5 .4) i;iO

On the other hand, it is easy to check that

M ( x, y)

=

x(x , y) . pn

Hence,

x( x, z) x( y, z) x( x, w) x( y, w) dev G 0

THEOREM

5.12. For any graph G, we have dev G < L (l - A; ) 4 + 20Virr G. ;;e o

The proof is essentially the same as the proof above, but somewhat messier. We will skip the proof here.

So, we see that small � and the condition of being almost regular together imply that the deviation of G is small. From Theorem 5. 10, small deviation in turn implies that for almost all x, y, N., n N11 is close to the expected size p 2 n. Since the count of ( labelled ) 4-cycles as subgraphs is exactly :E.,, 11 I N., n N11 12 , small dev G implies that this count is close to the expected number of such subgraphs in a random graph with the same edge density. While the deviation is very effective in dealing with graphs with large edge density, we point out here that for sparse graphs ( for example, graphs with no 4-cycles) , the deviation is not as useful. Still, for dense graphs, the deviation is quite powerful. For example, all of the quasi­ random properties in the list in Section 5 . 1 are implied by the deviation property since, in fact, dev G = o( 1 ) can also be added to the list of (equivalent ) quasi­ random properties. The interrelations of various quasi-random properties will be discussed in the next section.

5.4. Quasi-random graphs To simplify the discussion, we assume that all graphs G in this section are almost regular, i.e., .

zrr G

p

= -

"' £.....

(- 1

n vE V(G) Pv

1 ) p

-

=

o(1 )

where Pv = dv /n, p = L dv fn 2 and I V ( G ) I = n. Our goal here is to derive equivalence relations between various graph invariants and the eigenvalue property:

5.

82

EIGENVALUES AND QUASI-RANDOMNESS

X = m ax l 1 i#O

-

.A; I

=

o(1)

where .A; 's are eigenvalues of the Laplacian of G. Some directions of these relations have already been proved in the previous sections. Here we first consider bounding the discrepancy of a graph by X by using Theorem 5 . 1 . THEOREM

where X

=

5 . 1 3 . For a graph G , the discrepancy of G satisfies

maxi#o 1 1

-

disc G � ( X + 2Virr G)vol G

>. ; I -

To prove Theorem 5.13, we need to say a little more about quantities associated with being almost regular. As we recall, the irregularity irr G of G was defined in connection with the proofs involving deviation. A simpler definition could be the variance 1 L: 2 var G = p2 n v

(Pv - P)

·

Ostensibly, the variance var G goes to zero if and only if irr G does. It would simplify our discussion if these two quantities were "equivalent" in the sense that "v ar G � c1 irr G" and "irr G � c2 var G" for some constants c 1 and c2 • However only the first of these two relations is true (Sn , a star on n vertices serves as a example. ) In addition , the statement of Theorem 5 . 1 2 will not hold if we replace irr G by var G. This dictates then our choice of taking the upper bound as irr G. LEMMA

5 . 1 4 . For positive reals a1 , . . . , a n with average a, we have n 1 a n 1 1 2 L 2 ( a; - a ) � � 2 �). a n '

L(;- -

i=l

i=l

The proof is straightforward . Consequently, we have var G �

irr G.

In fact, for any subset X of V(G) , we have (5.5 )

var G

>

p n L...J

vEX

>

( L (Pv - )

-12 " (Pv p2 n � X I

p) 2

vEX

2

p)

(vol X - pn iX I ) 2 p2 n/l X I Before we p roceed to prove Theorem 5 . 13, we remark that there are two possible ways that we could define discrepancy: =

sup le(X , Y) - p i X I I Y I I X , Y<;; V(G)

5 . 4 . QUASI-RANDOM GRAPHS

83

or sup X £:;; V ( G )

\e(X, X ) - p\X\2 \.

These two quantities are bounded from each other to within a constant factor since

e(X U Y, X U Y) e(X, X ) + e(Y, Y ) + 2 e ( X, Y ) . =

So we will use whichever version that i s convenient for the proof, always allowing for a constant factor.

Proof of Theorem 5. 1 3:

From Theorem 5.2, we have (vol X? vol G

\e(X , X) Therefore,

disc G

sup X £:;; V(G) <

sup

X £:;; V ( G)

- X. I � .Xvol

\e(X, X ) - p\X\2\ \ e(X, X)

-

X) 2 ( v ol X) 2 1 + \ (vol vol G vol G

- p\X\ 2 \

From Lemma 5 . 7 we have

X - pn\X\) (vol X pn2 2(vol X - pn\X\) 2Jvar G · pn2 2Virr G pn2 . (vol

< < <

+ pn\X\)

·

The proof of Theorem 5 . 13 is complete.

0

We remark that we can replace irr G by var G in the statement of Theorem 5 . 1 3 by using ( 5 . 5 ) . Theorem 5 . 1 3 indicates that the eigenvalue property P. and the almost regular property imply the discrepancy property:

disc G In other words, P.

+

Pr

=>

=

o (vol G) .

Pdis c ·

We will denote the deviation property

dev G

=

as

follows:

o(1).

From Theorem 5 . 1 2 , we have

\dev G - L: (l - .X;) 4 \ i;tO

� 20Virr G.

5. EIGENVALUES AND QUASI-RANDOMNESS

84

As a consequence of the above inequalities and the fact that

.X4

�

�)1 - .\; ) 4 ,

i ;,.!O we have PdetJ + Pr =? Pe . To complete the equivalence of the three properties Pe + Pr , Pdi • c + Pr , PdetJ + Pn we will prove T HE O REM

5. 15. For a graph G, we have disc G � pn2 ( dev G) 1 14 ,

dev G � p44n2 disc G + 2 var G � p44n 2 disc G + 2 irr P RO O F .

G.

X, Y C V, p4n4dev G = L x(x, y )x( y , z)x(z, w)x(w, x ) z , y ,z, w = L: (L: x(x , y ) x(y, z)) 2 x,z y > L < E x(x, y ) x( y , z)) 2 z,zEX y > L L x(x, y ) x( y , z )) 2 1 1 2 ( z,zEX y 1 "" "" � I 2 (L.) L.. x(x, y )) 2 ) 2 XJ y zEX � 1 1 < L: < L: x(x , y)) 2 ) 2 2 yEY zEX 1 > 4 JXJ 2 J Y J 2 ( L L x(x , y))

First we consider, for

;

;

yE Y z E X

1 4 2 JXJ j Y j 2 ( e ( X , y ) - p J XJ J YJ )

This implies

For the second inequality, we consider the deviation of G using Theorem

p4n4dev G L (INz n Ny J - /n) 2 . z ,y

5.10:

=

For a fixed

x , we consider

Wz = L ( JNz n Ny j - p2n)2• y We note that JNx n Ny J is exactly the number d� = je (y , Nz) l of edges from y to the set Nz . We consider the set X consisting of the vertices y with d� � pdx . We have

L Jd� - pdz l =J e ( Nz , X) - pdz J X J J � disc G. yE X

5.4.

QUASI-RANDOM GRAPHS

85

Similarly,

L ld� - pd.,j = I e(N., , X ) - pd., I X I

I S disc G

z�X

Therefore

L(d� - d.,p) 2 $ 2n disc G II

and II

< <

L 2 {jN., n Nil I - p2n) 2 + 2n(pd., - p2n)2 4n disc G + 2p2 n 3 (p ., - p) 2 • II

Hence

p 4 n 4 de v G

<

L W., $ 4n2disc G + 2p2n2var G. "'

0

We have so far proved the equivalence of the discrepancy, deviation, and eigen­ value properties for almost regular graphs. In fact, all of the properties P1 , P2 , P3 , P4 have been covered for graphs with edge density 1/2. As for the property P5 (s) , for any graph M(s) on s vertices, s � 4, the number N(; (M(s)) of occurrences of labelled induced subgraphs of G isomorphic to M ( s) satisfies

N(; (M(s))

=

(1 + o( 1 ) )n" 2 - ( ; ) .

The proof of the equivalence of P5 (s) , for s � 4, with the other quasi-random properties is somewhat complicated and can be found in [76] . A quantitative version was proved in [74] . The difference of N(;(M(s)) and its expected value can be bounded by a function of the deviation of G. Namely, for a graph G with edge density 1/2, we have

This property is quite strong in the sense that quasi-randomness implies that if • a graph G with edge density 1 /2 contains ( 1 + o(1) ) 76 C4 's as labelled subgraphs, then G contains the expected number of Petersen graphs ( as well as any graphs of a fixed size ) . On the other hand, there are examples of graphs containing the correct number of all subgraphs on 3 vertices but which nonetheless fail to be quasi-random.

Notes: Q uasi-randomness resides at the intersection of extremal graph theory and ran­ dom graph theory. The original paper [76] ( also see [72 , 224] ) on quasi-random graphs provides a general framework for examining the relationships of combina­ torial structures as well. There are several papers on quasi-random hypergraphs [67, 70, 74, 77] and their connection with cohomology [78] and communication complexity (88] . Work has also been done on quasi-random sequences , subsets

86

5. EIGENVALUES AND QUASI-RANDOMNESS

(75] and tournaments (7 1 ] . Many questions on the quantitative analysis of the o( ·) terms remain open.

CHAPTER

6

Expanders and explicit constructions

6 . 1 . Probabilistic methods versus explicit constructions Many problems in combinatorics, theoretical computer science and communi­ cation theory can be solved by the following probabilistic approach: To prove the existence of some desired object, first an appropriate (probability) measure is de­ fined on the class of objects; second, the subclass of desired objects in question are shown to have positive measure. This implies that the desired objects must exist. This technique, while extremely powerful, suffers from a serious drawback. Namely, it gives no information about how one might actually go about explicitly constructing the desired objects. Thus, while we might even be able to conclude that almost all of our objects have the desired property (that is, all except for a set of measure zero) , we may be unable to exhibit a single one. A simple example of this phenomenon from number theory is that of a normal number. A real number x is said to be normal if for each integer b � 2, each of the digits 0, 1 , , b - 1 occurs asymptotically equally often in the base b expansion of x. It is known that almost all (in Lebesgue measure) real numbers are normal, but no one has yet succeeded in proving that any particular number (such as 1r, e or J2) is normal. ·

· ·

One of the earliest examples of the above probabilistic method is Erdos' clas­ sical result (1 1 1] from 1947 on the existence of graphs on n vertices which have maximum cliques and independent sets of size 2 log n (see Chapter 5, Section 2). Since then, probabilistic methods have been successfully used in a wide range of areas. However, in spite of the success of probabilistic methods, there is a clear need for explicit constructions, especially for applications in algorithmic design and in building efficient communication networks. In the past ten years, substantial progress has been made on explicit construc­ tions of so-called expander graphs. Although the developments in constructing expander graphs were first motivated by applications in communication networks, it has become one of the most powerful tools in complexity theory, parallel ar­ chitectures, distributed computing, derandomized algorithm, and the list goes on (30, 1 20, 205] . Still, we have not yet rigorously defined "explicit constructions" versus "random graphs." So, some explanations are in order. When we mention random graphs, some probability distributions are involved. For example, a typical model for graphs on n vertices is to assign equal probability to each of the 2 (; ) possible graphs on n (labelled) vertices. We say a random graph G on n vertices satisfies property P if all except for o( l) of the graphs satisfy P where the o( 1 ) term goes to zero as 87

88

6 . EXPANDERS AND EXPLICIT CONSTRUCTIONS

n approaches infinity. On the other hand, by an explicit construction, we mean a deterministic description from which we can construct a graph on n vertices with the number of steps polynomial in n, for infinitely many n . In other words, an explicit construction is a polynomial time algorithm for defining a graph. In fact, all of the explicit constructions we describe have very short descriptions. This, of course, is just one possible interpretation of "explicit construction" .

What is an expander graph? Roughly speaking, in an expander graph, any "small" subset of vertices has a relatively "large" neighborhood. We will leave the detailed description of the quantitative definitions of "small" and "large" until the next section. Conceivably, an expander graph can allow us to build networks with guaranteed access for making connections or routing messages. The key to the success in constructing expander graphs is its relationship with eigenvalues . Such a relationship can be used in two ways. Validating a random construction. One of the shortfalls of the probabilistic method is that it only guarantees the existence of an expander. A random graph, which is usually generated by using a random source (e.g. flipping a coin, etc.) , has high probability being an expander. Since eigenvalues can be computed easily (in polynomial time) , we can validate the randomly generated graph by computing its eigenvalues and the associated expander constants. While the probabilistic method does not guarantee the gen­ eration of expanders, it assures us that with high probability (say, within a few tries) we can obtain graphs whose performance in expansion is backed up by the eigenvalue bounds. 1.

2. Constructing large families of expanders.

A general eigenvalue lower bound can often be used to establish a whole family of expander graphs. In particular, algebraic and number-theoretic techniques can be used to bound the eigenvalues for special (infinite) families of graphs.

In the next section we will discuss various notions of expansion properties and their relations to eigenvalues. A variety of examples and explicit constructions will be given including Paley graphs, coset graphs, Margulis graphs, and Ramanujan graphs . In Section 6.4, we describe some classical applications of expanders to communication networks and we illustrate how to use expander graphs to create various building blocks for these networks. In Section 6.5, we discuss constructions for graphs with small diameter and girth. In the last section, we consider an extremal invariant of a graph which depends on the maximum eigenvalues of all possible weighted Laplacians. This invariant is closely related to Lovasz's {} function and it can be computed in polynomial time. We will describe its relationship to the chromatic number and clique number of a graph.

6.2. The expanders The expansion property is crucial in many applications [24, 172, 180, 208, 209, 206, 237, 242) and has become a driving force for recent progress in construc­ tive methods. Success here is due, in large part, to a combination of tools from graph

6.2. THE EXPANDERS

89

theory, network theory, theoretical computer science and various mathematical ar­ eas such as number theory, representation theory, and harmonic analysis. Perhaps, because of the large number of different applications in disparate settings, the def­ initions of expansion-like properties vary from one situation to another, often with a variety of names such as expander, magnifier, enlarger, generalizer, concentrator, and superconcentrator, just to name a few. To make matters worse, most of these definitions involve a large number of parameters. One typical example, namely the definition of a concentrator, is as follows: An (n, 9 , k, a , ,B)-concentrator is a bipar­ tite graph with n inputs, On outputs and k n edges, such that every input subset A with I A I � a n has at least .B n neighbors. It is conceivable that such tedious definitions actually hindered early progress in this area. The expansion property basically means each subset X of vertices must have "many" neighbors. That is, the neighborhood set N(X) = {y : y is adjacent to some x E X} is "large" in comparison to the size of X. The difficulty lies in finding a good way to define "many" or "large," quantitatively. There is an obvious scenario in which the subset S is almost the entire vertex set and the vertex boundary 8(S) = N(S) - S is very small. The typical definition of expander graphs is as follows: A regular graph G on m vertices is a c-expander if every subset S of V (G) satisfies

Here the constant c is often called the expander coefficient. Another way to define the expansion factor is as follows: I

c

=

18 (S) I S s�V(G) l I I S I /n sup

·

We remark that this definition is somewhat unsatisfactory for certain problems since the expansion factor c1 and the degree k are intimately related. For example, a random regular graph of degree k has an expansion factor of roughly k when the subset is small. The values of c and c1 should be judged in comparison with a function of k. In such cases, it works out better to formulate the lower bound for I 8(S) I directly in terms of eigenvalues. The expansion factor c1 is closely related to the Cheeger constant. In particular, for a regular graph G ,

c where ga

:=

.

mfs

vol 8S min(vol S, val

S)

I

1 > -cI - ga 2

>

, as defined

.

m

Chapter 2 .

The expansion of G is closely related to the discrepancy of G in the follow­ i ng sense: The discrepancy property implies every subset S contains close to the expected number of edges; therefore there are "many" edges leaving S.

6. EXPANDERS AND EXPLICIT CONSTRUCTIONS

90

Another related invariant is the isoperimetric number [45, 197] , denoted by i(G) , and defined by . ( G)

t

=

I { {u, v} E E(G) : u E S, v ¢ S} I

. min

IsI

S CN,I SI < -I

The so-called conductance is 1/k times i(G) for a k-regular graph G [166] . Clearly, for a k-regular graph, we have also i (G ) � k/2 - 2 disc( G) . We will give simple proofs which establish eigenvalue upper bounds for the expansion coefficient. Although the problem of checking whether a graph is an (n , k, c)-expander is co-NP-complete [29] ,the following relationship provides an ef­ ficient method to estimate the expansion and discrepancy of a graph. In Section 3.2, a similar bound was derived by examining the relation between diameter and eigenvalues. LEMMA

S

:

x

"'

y

6 . 1 . For a graph G and S � V (G) , the vertex boundary 6(S)

E S } satisfies

=

{x ¢

vol S vol G > .X ) vol S 1 .X (2 vol G vol S > .X (2 .X) vol G 2 where A = -Xt /(-X n - l + .XI ) . In other words, G is a .X(2 .A ) -expander. .X(2

vol 6(8) vol S

A)

_

_

_

_

-

PROOF . It follows from Theorem 3 . 1 and Lemma 3.4 of Chapter 3 that 1 - ( 1 - .X)2 vol 6S > vol S (1 - .X) 2 + vol Sfvol -

s·

By straightforward calculation, we get vol 6(S) vol S

>

( 1 - .X)2vol S + vol S .A ( 2 .X) vol S vol G vol S 1 .X(2 .X) vol G vol S A(2 .X) vol G _

=

_

>

_

_

as desired.

0

For a graph G and S � V (G) , we recall that the neighb orhood N(S) of a subset S is defined as follows: N(S)

=

{x

: x "'

y E

S} .

Note that S is not necessarily contained in N (S) . We consider the following varia­ tions of the expander lower bounds:

6. 2 .

THE EXPANDERS

91

L EMMA 6.2. Suppose G is not a complete graph. For S � V(G ) , the neighbor­ hood N(S) satisfies

vol N(S)

---:vol S� >

X2

1 +

(1

X2 ) vol S vol G

_

1 = -----� vol S 1 (1 vol G

X2 )

_

_

PROOF. In the proof of Theorem 3.1, suppose that we choose Pt ( .C) to be I and Y = V(G) - N(X) . Then we have O

=

>

.C

(Tlf2tjJy,pt (.C) Tlf 2 ¢x)

-____ �__,____ Vr-vol X vol � X vol Y vol � vol X vol Y y X vol G vol G

·

Thus X2 vol X vol

For Y = N(X) , this implies vol N(S) vol S

>

)2

1 +

{1

_

Y > vol X vol Y.

vo l S )2 ) vol G

=

1 1

_

(1

_

vol S )2 ) vol G

·

0

For regular graphs, we give a direct proof here. LEMMA 6 . 3 . Suppose G is a regular graph on n vertices and G is not a complete gmph. For S � V(G) , the neighborhood N(S) satisfies

where X =

I N(S) I > lSI max; ;eo

1

_x2 + ( 1 - .X2 ) W

=

1 (1 - ( 1 -

1 1 - .X; I .

PROOF. For S

c

V(G), we consider the characteristic function '1/Js :

'1/Js ( x) =

{ 01

if x E S otherwise

We consider the following inner product :

< (6.1 )

.X2 ) ��: �

a� + ( L: anX2 i2: 1

92

6. EXPANDERS AND EXPLICIT CONSTRUCTIONS

On the other hand,

(1/JsA, A?/Js )

=

=

L L I { w : {v, w} E E and {u, w } E E} I

u E S v ES

L I N (w) n S 1 2 . wEV

Applying the Cauchy-Schwarz inequality, we have:

L I N (w) n S I 2 wEV

>

CL:wEV I N (w) n s 1 ) 2 I N ( S) I

=

Combining this with (6. 1 ) , we obtain (6.2)

I

N ( S) I

lSI

>

0 For a bipartite graph , we can have a modified version for the expander theorem. For a bipartite graph G with vertex set X U Y and edges between X and Y , the incidence matrix M = M (G) has columns indexed by vertices in X and rows indexed by vertices in Y. For x E X and y E Y, the matrix M satisfies M ( x, y) = 1 if and only if { x, y} is an edge. A bipartite expander graph depends on the eigenvalues of M* M as follows: LEMMA 6 . 4 . For a bipartite graph G with vertex set X U Y and edges between and Y, suppose all vertices in X have the same degree. For a subset S of X, the neighborhood N(S) satisfies:

X

1 (1

_

p2 )

vol vol

S G

where p2 k2 is the second largest eigenvalue of M * M where bipartite adjacency matrix defined above.

M

M (G)

is the

P RO O F . Suppose every vertex in X has degree k. Let p; denote the eigenvalues of M* M . We denote the largest eigenvalue of M* M by P5 = k 2 • Let the a; 's denote the Fourier coefficients of characteristic function 1/Js with respect to the eigenfunctions of M * M. We consider the following inner product :

(6.3 )

:2 (1/Js M * , M l/Js)

=

<

I: a; p; ( I - p2 ) 152 1 n

+

P2 ISI.

6 . 3 . EXAMPLES OF EXPLICIT CONSTRUCTIONS

93

Together with

L L I { w : {v, w } E E and {u, w} E E} I

('1/Js M * , M 'f/;s)

uES vES

L I N (w ) n S 1 2

=

Combining this with

wE Y

I N ( w ) n s 1) 2 I N (S) I

>

CL:wEV

=

I SI 2k2 I N ( S) I '

{6.3), we have I

N ( S) I ->

lSI . {1 - p2 ) W + p2 D

We remark that if we consider the following modified matrix M for a bipartite graph ( not necessarily regular) , then the above proof works in a similar way. We define 1

M (u, v)

X

=

IT

y dy LEMMA 6 . 5 . For a bipartite graph G with vertex set X U Y and edges between and Y, and for a subset S of X, the neighborhood N (S) satisfies: 1 I N ( S) I > =

where

p2

lSI

- p2 + (1 - p2 ) W

1

(1 - p2) ��/ �

is the second largest eigenvalue of M * M .

We remark that the inequalities in Lemmas 6.4 and 6.5 are strict if the eigen­ values except for the largest are not all equal.

6.3. Examples of explicit constructions We will describe several useful families of expander graphs. For each construc­ tion, the precise value of a bound for the eigenvalues will be discussed. Together with the theorems in Section 6.2, these constructions yield good expanders. We will begin with graphs with edge density about ! and then proceed to graphs with lower edge density, say � for fixed k.

Construction 6.3. 1 . The Paley graph Pp . Let p be a prime number congruent to 1 modulo 4. The Paley graph consists of p vertices , 0, 1 , 2 , , p - 1 . Two vertices i and j and adjacent if and only if i - j is a non-zero quadratic residue modulo p. The eigenvalues of Pp are exactly · ·

·

1 - L.:

x E Z;

/��·· f (p -

1)

6. EXPANDERS AND EXPLICIT CONSTRUCTIONS

94

for each j = 0, , p - 1 . These are closely related to Gauss sums modulo p (see [165] ) . In particular, it is known that for any j "I 0 (mod p) , the Gauss sum · · ·

E., E z; e

2•ijz2 P

is either ..;P - 1 or - ..;P - 1.

Therefore, we have

.X1

=

Ap - 1

=

..;P - 1

12 (p - 1 ) ' 1

+

y'P + 1

2(p - 1) '

The Paley graph is a favorite and frequently cited example in extremal graph theory, no doubt due to its many nice properties the existence of which are guar­ anteed by eigenvalue bounds . For example, Pp contains all induced subgraphs on cJlog p vertices (see [35] ; also implicit in [146] ) .

Construction 6.3.2. The Paley sum graphs PP . The Paley sum graphs are basically the symmetric versions of Paley graphs without the constraint p = 1 (mod 4) . Let p be any prime number . PP has vertices 0, · · · , p 1 , and two vertices i and j are adjacent if and only if i + j is a quadratic residue modulo p. The eigenvectors for Paley graphs are ¢1 where ¢1 (k) = e 2"ii k , for j = 0 , . . . , n - 1 . For sum graphs, the eigenvectors are

where Aj is the eigenvalue for the eigenvector ¢J of the Paley graph. Therefore the eigenvalues for the sum graphs are exactly 1 ± ../1 ( 1 - AJ ) ( 1 - L i ) J . For p = 1 (mod 4 ) , the eigenvalues of the Paley sum graph is the same as those of the Paley graph. For p = 3 (mod 4), the eigenvalues of the Paley sum graph are 0, 1 (with multiplicity

�) and 1 + 2 J;=r

(with multiplicity

� ).

2 J-=r

The Paley graphs and Paley sum graphs both have edge density about � · They can be generalized to graphs with edge density � for any fixed constants t and r with t < r. Paley sum graphs are actually a special case of the following:

Construction 6.3.3. The generalized Paley sum graphs Pp ,r,T · For a fixed integer r > 0, let p = mr + 1 be a prime congruent to 1 mod 4 and let T C z; consist of t non-zero residues so that for any distinct a, b E T, ab- 1 is not an rth power in z; . The generalized Paley graph has vertex set {0, 1, . . . , p - 1 } . Two vertices i and j are adjacent if and only if i + j = aq for some a E T and q an rth power. The eigenvalues are Aj = ( 1 - L L e2"ij a z 2 ) / (p - l ) t .

a E T zE Z;

For j 1= 0 , using the well-known theorem of Deligne [94] , we have J

• L e 2 rr ijz J :$ (

zEZp

r -

l)v'j}.

6.3.

EXAMPLES OF EXPLICIT CONSTRUCTIONS

95

Therefore the eigenvalues of the generalized Paley sum graphs Pp,r,t satisfy

,

-" 1 2: 1 -

, -" n - 1 $ 1 +

(r - 1) y'P + 1 , p-

1

(r - 1) y'P + 1 . p- 1

In the other direction, Paley graphs can be generalized to the following coset graphs on n vertices with edge density n- l + t for any positive integer t (see [5 1] ) .

Construction 6.3.4. The coset graphs

Cp , t .

We consider the finite field GF(pt) and a coset x + GF(p) for x E GF(pt) � GF(p) (x) . There is a natural correspondence between elements of the multiplicative group GF* (pt ) and elements of { 1 , pt - 1 } . For example, choosing a generator g, each element y in GF* (pt) corresponds to the integer k for which y = g k . Now we consider the coset graph Cp, with vertices 1 , , pt - 1 = n, and edges { a, b} t if a + b is in the subset X of integers corresponding to the coset x + GF(p) . The · ·

·

,

·

eigenvalues of the coset graph

1.

Cp, t

· ·

are :L aE X e a for () ranging over all nth roots of

Bounding the eigenvalues of coset graphs leads t o a natural generalization of Weil's character sum inequality [1 65] The following inequality was conjectured by the author [ 51] and proved by Katz [170] and others [181, 183] . If () is a (pt - 1 )-th root of 1 and () -:f 1 , we have .

I where X is the coset

L ()a 1 :5 (t - 1 h/P,

aEX

x + GF(p) .

The coset graph has edge density nI- t , and the eigenvalues satisfy >.1 2: 1 - t:J,

and An-I :5

1 + t:J, .

Construction 6.3.5. The Margulis graphs Mn . In the early 70's, Margulis [192] ignited an entire movement toward the study of constructive methods by relating Kazhdan 's property T to expanders . This approach was successfully continued by Gabber and Galil [137] who obtained ex­ plicit values for estimating the expander constant . Here we illustrate some of these early constructions of these very elegant graphs, which we call Margulis graphs [8, 137, 192] . Set n = m 2 and V = Zm x Zm . Consider the following six transfor­ mations from V to itself:

O"J (x, y) = (x, y + 2x) (x, y + 2x + 1 ) a2 (x, y) (x, y + 2x + 2) a3 (x, y) (x + 2y, y) a4 (x, y) (x + 2y + 1 , y) as (x, y) (x + 2y + 2, y) as (x, y) (Addition here is modulo m. ) Let

G

Mn

=

( V, E) be the graph with vertex set V and with edges { u, v} if (Thus, e.g., (0, 0) is joined to itself by 2 loops -

u = a; (v) or v = a; ( u ) for some i . =

6. EXPANDERS AND EXPLICIT CONSTRUCTIONS

96

note that here we consider that a loop adds 2 to the degree of a vertex.) Obviously, G is 1 2-regular. Claim:

2 - vfJ A 1 ? -3- .

PROOF. Let T be the ( 0 , 1) x ( 0 , 1 ) torus, and define two measure-preserving automorphisms 1/J 1 , 1/J2 on T by 1/J1 (x, y) = (x, y + 2x) , 1/J2 (x, y) = (x + 2y, y) , where addition is modulo 1 . The main part o f the proof i s based on the following fact proved as Lemma of [137) . Here we state without proof this analytic result: H ¢ is measurable on T and

£I¢

(6.4)

· 1/J! l - ¢

where c = 4 - v'i2.

4

IT ¢ = 0 , then

12 +

lr I ¢ · 1/J21 - ¢ 12 ? c lr ¢2,

Now suppose that f : V -t IR satisfies "LI:k= l f (j , k) = 0 . Define a measurable function ¢ : T -t IR as follows: If (j, k) E Zm x Zm then for . "+1 k k+1 ]_ 'S. x < 1-- , - 'S. y < -- , ¢ ( x , y ) = f (j , k ) . m

m

m

m

It can be checked that IT ¢ = 0 , and

<

�

2 2

Also IT ¢2 =

L

(u,v) EE

� Lv E V J 2 ( v ) .

�

2 2 Thus,

( f (u) - f (v) ) 2 .

(v,u)EE

Lf=0 The claim is proved.

L

T herefore, by

( 6.4 ) we have

( f (v) - f (u ) )2 ?

implies

L

(v,u) E E

�2 L jZ (v) . vE V

( f ( v ) - /(u))2 ? 2c(f, /} .

0

We can construct graphs with larger degrees and bounded eigenvalues by taking products of Mn as follows. The graph M� has vertex set V, and two vertices u

6.3.

EXAMPLES OF EXPLICIT CONSTRUCTIONS

97

and v are joined by s parallel edges where s is the number of walks of length k in Mn from v to u. Thus the adjacency matrix of M� has eigenvalues p� where p; are eigenvalues of the adjacency matrix of Mn · Although this construction does not give as good eigenvalues as the following Ramanujan graphs, the construction schemes are simple and the approach is interesting.

Construction 6.3.6. The Ramanujan graphs Xp,q . One of the major developments in constructive methods was the construction of Ramanujan graphs by Lubotzky, Phillips and Sarnak [191] and independently by Margulis [193, 194, 195] . Ramanujan graphs are k-regular graphs with A 1 satisfying

2

\ > 1 - v'k""=T 1\1 k .

..

This eigenvalue bound is the best possible for fixed k where the number n of vertices approaches infinity (see Section 2 4) The construction can be given for large classes of parameters for k and n using quotients of quaternion groups: : Let p be a prime congruent to 1 modulo 4 and let H(Z) denote the integral quaternions

H(Z) = { o = a0 + a 1 i + a2 j + a3 k : ai

E

Z} .

Let a = a0 - a 1 i - a 2 j - a 3 k and N (o) = oa = a� + a� + a� + a� . It can be shown that there are precisely � conjugate pairs { o, a} of elements of H ( Z) satisfying N(o) = p, o = l {mod 2) and a0 > 1. Denote by S the set of all such elements. For each o in S, we associate the 2 x 2 matrix a a =

( -aao2++i�tat 3 a2 +-i�tat3 ) ao

Let q be another prime congruent to 1 modulo 4. By taking the i in a to satisfy i 2 = - 1 ( mod q) , a can be viewed as an element in PGL(2, ZjqZ) , which is the group of all 2 x 2 matrices over ZjqZ. Now we form the Cayley graph of PG L(2, ZJ qZ) relative to the above p + 1 elements. (The Cayley graph of a group G relative to a symmetric set of elements S is the graph with vertex set G and edges { x, y} if x = sy for some s in S.) If the Legendre symbol = 1 , then this graph is not connected since the generators all lie in the index two subgroup PSL(2, ZjqZ) , each element of which has determinant a square. So there are two cases. The Ramanujan graph Xp,q is defined to be the above Cayley graph if = - 1 , and to be the Cayley graph of PSL(2, ZjqZ) relative to S if = 1.

( �)

(�)

(�)

(�)

For = - 1 , Xp, q is bipartite with edges between PSL(2, ZjqZ) and its set theoretic complement. The Ramanujan graphs of interest here correspond to taking = 1 and are (p + I ) -regular graphs with q { q2 - 1 ) / 2 vertices . The bipartite

( �)

graphs with ( £) q

=

- 1 are bipartite expander graphs.

�

The eigenvalue A t can be determined to be 1 by using the results of Eichler [109] and Igusa [163] on the Ramanujan conjecture [191, 214] for the above case. In addition to having the optimum eigenvalue A 1 , Ramanujan graphs

98

6. EXPANDERS AND EXPLICIT CONSTRUCTIONS

have many other nice properties. They serve as illuminating examples for various extremal problems, several of which we will discuss in Section 6.5.

6.4. Applications of expanders in communication networks There are many applications of expander graphs scattered about in recent re­ search papers in theoretical computer science. Expanders are extensively used in randomized and derandomized algorithms, computational complexity, and parallel architectures. There are many "tricks of the trade" in using expanders for am­ plifying a random bit, generating pseudorandom numbers , and constructing error­ correcting codes, for example. We do not intend to cover these new applications since there are many more still being developed. Instead, we will focus on a classical example of using the power of expanders in communication networks. Among various applications of expander graphs, those in communication net­ works have the longest history, and can be traced back to the early development of switching networks [63, 192, 208, 209] . Roughly speaking, a telephone network which provides connections for users is made of many parts or "gadgets" , each of which performs some desired function. We will describe some of the gadgets that can be built by using expanders . A non-blocking network is a directed graph with two specified disjoint subsets of vertices, one of which consists of input vertices and the other of output vertices. · Now suppose that a number of calls take place simultaneously in the network, i.e. , there are vertex-disjoint paths joining some inputs to outputs in the graph . Suppose one additional call comes in and it is desired to establish a new pat.h joining the given input to the desired output without disturbing the existing calls, i.e . , the new path is to be vertex-disjoint from the existing paths. The problem of interest is to minimize the number of edges in such a non-blocking network.

To build a non-blocking network, we need several types of building blocks , one of which is called a k- access graph, which has the property that for any given set S of vertex-disjoint paths connecting inputs to outputs, a new input can be connected to k different outputs by paths not containing any vertex in S. If k is greater than or equal to half of the total number of outputs, the k-access graph is a so-called major access network. A non-blocking network can then be formed by combining a major access network and its mirror image as shown in Fig. 1 . We construct here a major-access network M(n) with n inputs and 24n outputs by combining 2 copies of M (n/2) and 2 copies of the bipartite Ramanujan graphs R( 1 2n , 5) with 12n inputs and with degree p + 1 = 6, as illustrated in Fig. 2. To verify that the above construction is a major-access network, we consider an input v which must have access to 6n of the middle vertices . After deleting the n possible vertices in S, the remaining set has at least 5n inputs of M (n) . In each of the Ramanujan graphs with degree k = 6 and ..\ = 1 -./5/3 , we see that , by Lemma 6.4, the neighborhood of a subset S of size 5n has size at least : 5n 27n = ( 1 - ..\)2 + f2 ..\ (2 - ..\) 4" -

6 . 4 . APPLICATIONS OF EXPANDERS IN COMMUNICATION NETWORKS

n inputs

n

99

outputs

maj or-access network F IGURE

1 . A nonblocking network

!! 2

12n

!! 2

12n

M(n) F I G U RE

2 . A major access network

Since there are two copies of the Ramanujan graph R( 12n, 5) here, there are at leas t 2�n neighbors of S. Among the ¥-n such outputs, there are at least ¥-n of them not in S, which is more than half of the outputs of M (n) . Therefore the above construction yields M ( n) and the number of edges of M ( n) satisfies

e ( M (n) )

=

n

2e(M( - ) ) + 6 · 1 2 2n. z ·

It c an then be easily checked that the above major-access network has at most 144n log n edges and therefore the nonblocking network has at most 288n log n edges.

1 00

6. EXPANDERS A N D EXPLICIT CONSTRUCTIONS

Another useful network is the so-called superconcentrator. Despite this impres­ sive name, it actually has a very simple property. Namely, it is a graph with n inputs and n outputs, having the property that for any set of inputs and any set of outputs , a set of vertex-disjoint paths exists that join the inputs in a one-to-one fashion to the outputs (so that here it does not matter who is connected to whom.) The question of interest is to determine how few edges a superconcentrator can have. In fact, this has been taken as a measure for comparing the effectiveness of the expanders which are used to build superconcentrators. A simple recursive construction [192] of a superconcentrator is shown in Figure 3.

n output

n input

s c;) S(n) FIGURE 3 . A superconcentrator In the network in Figure 3, there is a matching between the n inputs and n outputs. Furthermore, the graph B has n inputs and 5n/6 outputs satisfying the property that for any given n/ 2 inputs there is a set of vertex-disjoint paths joining the inputs in a one-to-one fashion to different outputs . For example, as defined in Section 6. 2 , an (n, 5/6, k, 1 /2, 1 /2)-concentrator has the above property. So for any given set of m inputs and m outputs in S(n) of Figure 3, we can use the matching to provide m - n/ 2 disjoint paths and let the rest be achieved recursively in S( 5; ) Therefore the key part of the construction is made from an expander as in Figure 4. .

I n Figure 4, the first n / 6 inputs, each having degree 5 , are joined t o 5n/6 distinct outputs. The remaining 5n/6 inputs are joined to the outputs by a Ramanujan graph with degree 6 = p + 1 . Now suppose we have a set of inputs X. It suffices to show that X has at least I X I neighbors as outputs . Here we verify the situation for I X I= n/ 2 (the other cases where I X I < n/2 are easier) . If X contains at least fij inputs among the first � inputs, then we are done. We may assume X contains at least 25n inputs as an input set X' of the expander C. Since the expander graph has eigenvalue .A = 1 v'5/3, it is straightforward to check that -

N(X ' ) > - .A(2

I X' I

-

.A) 1},;1 6

+

(1

-

.A )2

�

9 - �n 4

125 + 5

. 25

> n/ 2 .

6.5. CONSTRUCTIONS OF G RAPHS WITH SMALL DIAMETER AND GIRT H

n input

c

Expander

101

5n/6 output

FIGURE 4 . A concentrator Now the total number of edges in the superconcentrator S (n} satisfies

e(S(n}}

= n + 2e(B) + e(S(5n/6} } = n + 2 5/6n 7 + e(S(5n/6}} ·

·

It is easy to verify that the above superconcentrator S(n) has at most 76n edges. The number of edges in S(n) can be reduced to 69.8n by replacing S(�n) by S( ( � + t: }n) where t: = .0288776, and in B each of the first ( � - l )n inputs of B has degree 4 or 5, and they are joined to a total of ( � + t:}n distinct outputs of B. I t i s worth mentioning that by using expanders, the existence o f which i s guar­ anteed by probabilistic methods, one can construct superconcentrators with as few as 36n edges. The parameters of such probabilistic expanders are given in Theorem 6.6 (also, see (17] } . We remark that he best currently known lower bound for a superconcentrator of size n is 5n + O (logn} , due to Lev and Valiant [ 1 8 2] .

THEOREM 6 . 6 . For real numbers 0 < a < 1 //3 < 1 , suppose that a n integer k satisfies

H (a) + H(a/3) H (a) - a{JH(1 / /3} where H(x) = -x log2 x - ( 1 - x} log., ( 1 - x ) is the entropy function. Then for any integer n, there exists a k-regular bipartite graph with vertex set A U B , I A I = I B I = n , so that every subset X s:;; A with l X I � an has at least fJ I X I neighbors in B . k

>

I n fact almost all random k-regular bipartite graphs between two sets o f n vertices satisfy the above property. The proof is by combinatorial probabilistic methods and can be found in (63] .

6 . 5. Constructions of graphs with small diameter and girth There are many related extremal properties that are satisfied by random graphs but are somewhat "weaker" than the properties mentioned in Section 6.2. One such example is the diameter, which is the maximum distance between pairs of vertices .

1 02

6.

EXPANDERS AND EXPLICIT CONSTRUCTIONS

There are graphs with small diameter but which do not have good expansion, discrepancy or eigenvalue properties. The following two related extremal problems often arise in interconnection net­ works [110] :

Problem 6. 5. 1 . Given k and D , construct a graph with as many vertices as possible with degree k and diameter D.

Problem 6. 5. 2. Given n and k, construct a graph with minimum degree on n vertices with diameter as small as possible .

It is not difficult to see that a graph with degree k and diameter D can have at most M ( k , D) = 1 + k + + k(k - 1 ) D-1 vertices, which + k(k - 1)i-1 + is sometimes called the Moore bound . Let n(k, D) denote the maximum number of vertices in a graph with degree at most k and diameter at most D. Clearly, n ( k, D) :::; M(k, D) . It has been shown that there are at most five kinds of graphs achieving the Moore bound, namely, cliques, odd cycles, the Petersen graph, the Hoffman-Singleton graph (with k = 7 and D = 2) and possibly the case of M(2, 57) (which remains still unresolved [113, 160] ) . · ·

·

·

· ·

A random graph h as small diameter . To b e specific, Bollobas and d e l a Vega [36] proved that a random k-regular graph has diameter logk _ 1 n + logk_ 1 log n + c for some small constant c < 10. This is almost the best possible in the sense that any k-regular graph has diameter at least logk _ 1 n. Upper bounds for the diameter in terms of eigenvalues were discussed in Chapter 3. Namely, a k-regular graph G . r log(n - 1 ) l on n vert1ces h as d"1ame ter at mos t tog(1/( J - >.)) .

Using the above bound, the Ramanujan graph has diameter at most f t s( �o/� �>.)) l l o which comes to within a factor of 2 of the optimum diameter. In other words, the Ramanujan graph has the number of vertices about a factor of (k - 1 ) - D/2 times the Moore bound [160] . Quite a few other constructions, such as de Bruijn graphs [42] and their variations, fall within the range of 2-D times the Moore bound. It remains an open problem to determine the maximum number n(k, D) of vertices in a graph with degree k and diameter D. Relatively little is known about the upper bound for n(k, D) . The following somewhat trivial sounding question concerning the upper bound is still unresolved [1 13] :

Problem 6. 5. 3. I s i t true that for every integer c , there exist k and D such that n(k, D) < M(k, D) - c? There are many papers on this problem; the reader is referred to [2 1 , 22, 46, 65, 66] for surveys on this topic. Another direction is to allow additional edges to minimize diameter:

Problem 6. 5.4. How small can the diameter be made by adding a matching to an n-cycle? It was shown in [32] that by adding a random matching to an n-cycle the resulting graph has best possible diameter in the range of log2 n. In fact, a more

6.6. WEIGHTED LAPLACIANS AND THE LOV Asz iJ FUNCTION

1 03

general theorem can be proved which ensures that by adding a random matching to k-regular graphs, say R.amanujan graphs, the resulting graphs have diameter about logk _ 1 n. It would be of interest to answer Problem 6.5.4 and its generalizations by explicit constructions [6, 18, 68, 1 33, 245] . Vijayan and Murty [245] asked the following problems related to reliability of graphs with diameter constraints:

Problem 6. 5. 5. Given n, D, D' and s , what is the minimum number of edges in a graph on n vertices of diameter D with the property that after removing s edges the remaining graph has diameter no more than D'? These problems have attracted the attention of many researchers. There is a large literature on this problem [21 , 22, 63, 65, 106] However, the answers are far from satisfactory and explicit solutions are few. Most problem of this type remain unsolved. The girth of a graph is the size of a smallest cycle in the graph [26, 164, 250] . The girth of a random k-regular graph i s known t o be logk _ 1 n [1 1 7] . In [19 1] , the bipartite R.amanujan graphs on n vertices have girth about � logk_ 1 n; which is better than that of a random graph in the sense of avoiding small cycles. (To be precise, the bipartite Ramanujan graph Xp,q has q(q2 I) vertices and girth 4 logP q - logP 4. ) This is closely related to the following extremal problem which, though rather old , remains open [37, 227] : -

Problem 6. 5. 6. For a given integer t 2:: 6, how many edges can a graph o n n vertices have without containing any cycle of length 2t? Erdos conjectures that the maximum number f(n, t) of edges in a graph on n vertices avoiding C2 t is of order n i + 1 ft . It is not hard to see that f(n, t) < n1 + 1 ft . The Ramanujan graphs yield f (n, t) > n1 +2f3t which is a substantial improvement upon previous lower bounds of nl + •• �. in [37] . Recently, Lazebnik, Ustimenko and Woldar [177] constructed graphs on n vertices, for infinitely many n, which have n 1 +2/(3t-3) edges containing no cycle of length 2t.

6.6. Weighted Laplacians and the Lovasz

t'J

function

For a simple graph G and a weight matrix W = (wu v ) where W uv = Wvu 2:: 0 and Wu v = 0 for u and v not adjacent or u = v, we define the W-weighted Laplacian .C w , a matrix with rows and columns indexed by vertices of G, 1 Wv,v if U = V , and Wv # 0 Wv

.C w (u, v)

=

-

Wuv JWuWv 0

where Wv

=

L Wuv . u

if u

�

v

and

otherwise,

u # v,

104

6 . E X PANDERS AND EXPLICIT CONSTRUCTIONS

Previously we have defined the Laplacian of a (fixed) weighted graph. The above definition is different in the sense that we start with a fixed graph G and consider all possible weight matrices. For a graph G with a weight matrix W, let >- maz (Gw ) denote the maximum eigenvalue of the W-weighted Laplacian of G. In particular, let >-maz (G) denote >-ma z ( G w) when the weight matrix satisfies W uv = 1 if u ,...., v and 0 otherwise.

The

a

function of a graph a

( G)

G is defined as follows: =

1+

1

nw' >-ma z ( G w ) - 1

where W ranges over all weight matrices.

The chromatic number x(G) of a graph G is the smallest integer k such that the vertices of G can be k-colored so that any two adjacent vertices have different colors. Bounds are known for x(G) in terms of the eigenvalues of the adjacency matrix A = (Au v ) of G ( where A uv = 1 if u and v are adjacent, and 0 otherwise ) . Let Pmin and Pmax denote the smallest and largest eigenvalues of A, respectively. An upper bound of x (G) was first given by H. S. Wilf [248] who showed that

x ( G) $ 1 + Pmax ·

For the lower bound , A. J . Hoffman [161) proved ( also see [153)}

x( G) � 1

_

Pmax Pmin

.

Here we will show that the a function serves as a lower bound for the chromatic number x (G) with a proof very similar to that in [161) .

THEOREM 6 . 7 .

For any graph G which contains at least one edge, we have 1

x(G) � 1 + m� >- ma x (Gw ) - 1 = a (G)

M1 with the ( u , v ) -entry to be ,jwuwv . T 1 121 as a row vector and ( T1 12 1}* as a column vector, we can write ( Tl/ 2 1 ) ( Tl f2 1 ) * = rt f2 JT tf 2. Mt where w = L W v and J is the all 1 's matrix. PROOF. We consider the matrix

regard

If we

=

v

We consider eigenvalues of

>. M = £ - 1 + - Mt w where >. = >-max ( G w ) denotes the maximum eigenvalue of the W -weighted Lapla­ cian £ of G. I t is easy to see that the eigenfunctions of £ are eigenfunctions of M and the maximum eigenvalue of M is >. - 1 .

6 . 6 . WEIGHTED LAPLACIANS AND THE LOY Asz 11 FUNCTION

105

Let X denote an independent subset of G and let Mx denote the principal submatrix of M with rows and columns restricted to vertices in X. Then

Mx = � ( M1 ) x = � ( T 1 / 2 l)x ( T 1 /2 1 ) x . w w The maximum eigenvalue of M is � (T 1 12t ) x ( T1 1 2 t ) = � w(X ) w

where

w

w(X) = L Wv · vEX

Since the maximum eigenvalue of a principal submatrix of a symmetric matrix M is no more than the maximum eigenvalue of M, we have

.X . .X - 1 � -w(X) w Suppose G has chromatic number x(G) = k. Thus , the vertex set of G can be partitioned into k independent sets X1 , . . . , Xk . We then have

.X - 1 �

- w (Xi ) , i w

.X

=

1, . . . 1 k .

In particular the sum over all i satisfies 1

k(.X - 1)

.X

>

- 'L: w (Xi ) w .

A.

=

This implies

x(G ) =

k � 1 + .x _ 1 1

and (5. 1 ) is proved.

D

As an immediate consequence of Theorem 6.71 we have the following corollaries:

COROLLARY 6 . 8 .

For any graph G, we have x (G) � 1 + mcpc: G

\

-"maz

1 (G'W ) - 1

where G' ranges over all induced su.bgraphs of G and m atrices for G' .

W

ranges over all weight

We note that the inequality above is strict for some graphs ( i.e. , odd cycles for example ) . We also have the following simpler and weaker inequality.

COROLLARY

6 . 9 . For a graph G, we have x (G) � 1

+

1 >.maz ( G) - 1

wh ere G' ranges over all induced subgraphs of G.

6. EXPANDERS AND EXPLICIT CONSTRUCTIONS

106

In a graph G, the clique number w( G ) of G is the number of vertices in G forms the largest complete subgraph (which is also called a clique). The problem of determining the clique number of a graph is known to be NP-complete [140] . A graph G is perfect if G and all its induced subgraphs have the property that the chromatic number equals the size of a largest clique. There is an excellent survey by Lovasz [189] on perfect graphs. In general, the clique number and the chromatic number can be quite different. Here we will show that the u function serves as an upper bound for the clique number (also see [221] ) .

THEOREM

6 . 10.

w(G) $ u (G)

(6.5)

Suppose the clique number of G is k. Clearly, k > 1 . Let S = { v 1 , · · · , Vk } denote the vertex set of the maximum clique. For each edge { u, v } , u, v E S, we define W u v = Wvu = 1 · Let () =f 1 denote a k-th root of unity, say, 2?Ti () = exp( y ) . P ROO F .

We consider f ( v)

=

()i for v i n Ci , 1 $ j $ k. Then we have

L I ( J ( u ) - f(v) Wwuv v #m

=

v

i

=

m

k(k - 1 )

k2 k(k 1 )

=

-

k

=

k- 1

Therefore w

and (6.5) is proved.

( G)

=

.

k$ 1+

.X

1 ma:z:

-1

Combining ( 5 . 1 ) and (6.5 ) , we have proved:

$ u ( G)

w(G) $ u (G) $ x(G) .

0

6.6.

WEIGHTED LAPLACIANS AND THE LOY Asz

6

107

FUNCTION

The a function is closely related to Lovasz's {) function which was first intro­ duced in Lovasz's seminal work on the Shannon capacity [188] . The {) function has many alternative definitions (see [151 , 152] } . One of the formulations states iJ(G)

W ') = 1 + max Pmaz (W Pmin( ') W'

where Pm a z ( W' ) denotes the maximum eigenvalue of W' and W' ranges over all matrices with rows and columns indexed by the vertex set of G satisfying W' ( u, v ) :f. 0 only if the vertices u and v are non-adjacent . The original definition for iJ(G) is as follows: For a graph G, an orthonormal is an assignment of a unit vector X v in a Euclidean space to each vertex v of G, in such a way that ( x u , x v ) = 0 if u :f. v and u is not adjacent to v . The {) function iJ( G) is equal to:

labeling of G

iJ(G)

=

1

min max z ,y v (Xv , Y ) 2

where the minimum ranges over all orthonormal labelings

x

and all unit vectors

y.

As pointed out by A . Galtman [138] , the a function can be expressed as : (6.6)

a( G)

W) = 1 + max Pmaz (W Pmin( ) W

where W ranges over all matrices with rows and columns indexed by the vertex set of G satisfying W ( u, v ) > 0 only if the vertices u and v are adjacent . Therefore, we have

a (G) :::;

iJ( G ) .

The above formulation for a (G) coincides with that of a function introduced by McEliece, Rodemich and Rumsey [1 98] in their study of bounds for the indepen­ dence number (The notation O: L was used there) . They showed that for a family of graphs the a function is equal to the Delsarte linear programming bo u nd. Inde­ pendently, A. Schrijver [221] considered the a function (with the notation {)' ) . In [221] , an example by M.R. Best was given for a graph H satisfying a(H) :f. iJ(H) . The vertex set of H is { 0 , 1} 6 and two vertices are adjacent if and only if they differ in at most three of the six coordinates. The following definition for {)'(G) = a( G ) was given by Schrijver: For a graph G, an acute orthonormal labeling of G is an assignment of a unit vector x v in a Euclidean space to each vertex v of G , in such a way that (xu , xv) � 0 if u = v or u is adjacent to v and 0 otherwise. Then 1 )2 a(G) = {)I (G) = mm max ( •

x ,y

v Xv 1 Y

---

where the minimum ranges over all acute orthonormal labelings vectors y . So, altogether we have the so-called

w( G)

Sandwich Theorem: :::; a( G) :::; 19( G ) :::; x( G ) .

x

and all unit

6 . EXPANDERS AND EXPLICIT CONSTRUCTIONS

108

How good are these inequalities? In other words, how effective can t?(G} or a(G} be used to estimate w(G} or x(G}? Although the experimental evidence indicates that they are good in many cases, theoretical results show that significant obstacles are present in this approach. Erdos [ 1 1 2] proved that

(6.7)

x (G} < -::-en --:-,:-; (log n)2 w (G} --

and for every n , there is a graph on n vertices satisfying c'n . x(G} > (6·8} w(G} - (log n)2

In fact, (6.8} is achieved by random graphs since Ramsey theory implies that a random graph has w( G) � 2 log n / log 2 and therefore its chromatic number x( G) � n/w (G) � c'n/ log n . It h as been shown ([124] , also s ee ( 1 3 , 122] ) that for some f > 0, i t i s impos­ sible to approximate in polynomial time the independence number of a graph on n vertices within a factor of n < , assuming P =I N P. The exponent f has recently been improved [157] , under similar assumptions, to 1 - fJ for every positive fl . On the positive side, there have been a number of recent papers o n approx­ imation algorithms for coloring problems, max-cut problems and clique numbers using the fJ function [11 , 154, 141, 173] . Although these approximation algo­ rithms often have complicated statements for certain ranges of the parameters with incremental or weak improvements, this approach still represents a significant step towards the understanding of these hard problems .

Notes:

The first four sections in this chapter are mainly based on a previous survey article [69] . Section 6 . 5 deals with several problems previously discussed in [65 , 66, 68, 32] . Concerning the last section on the a function and the chromatic number, the reader is referred to an excellent survey paper by Knuth [1 70] . Related material can also be found in [1 5 1 , 1 52]. The alternative definitions of the a function and its relationship to the rJ functions are given in Galtman [1 38] .

CHAPTER 7

Eigenvalues of symmetrical graphs

7. 1 . Symmetrical graphs Roughly speaking, graphs with symmetry have "large" automorphism groups . There are various notions of symmetry in graphs such as vertex-transitivity, edge­ transitivity, and distance transitivity (all of which we will soon define precisely) . In general, the eigenvalues of symmetrical graphs have many remarkable properties , several of which we now mention:

1 : For symmetrical graphs, eigenvalues can be easily bounded in terms of the diameter D and the degree k. For example, for vertex transitive graphs we have (7. 1) This implication has very interesting and important consequences . In many random walk problems, we might not even know the number of vertices in a graph, but the above inequality provides an efficient way to estimate the eigenvalues of a graph as long as the diameter and degree can be computed or bounded . This is a main component for the recent developments in rapidly mixing Markov chains [225] . Several eigenvalue inequalities with a form similar to ( 7. 1 ) will be derived in a clean way in Section 7.2.

2 : A graph is distance transitive if for any two pairs of vertices ( x , y) ,

(z, w ) with there is a graph automorphism mapping x to z and y to w . Distance transitive graphs have very special spectra. We will describe a simple and natural method to determine the spectrum of a distance transitive graph. Basically, the spectrum of a distance transitive graph is the same as that of a corresponding contracted path with D + 1 vertices where D is the diameter. In fact, a distance transitive graph has exactly D + 1 distinct eigenvalues . d( x, y) = d( z, w ) ,

3: For some symmetrical graphs , eigenvalues can be explicitly determined by using group representations. We will discuss the general methods and give two examples: i ntersection graphs and the "Buckyball" graph (the truncated icosahedron) . Inter­ s ection graphs have vertices labelled by subsets and adjacency determined by the cardi nality of intersections of the sets corresponding to the endpoints. The Bucky­ ball graph is associated with the newly found molecule "Buckminsterfullerene C5 o " , the third form of solid carbon. Not only does it have beautiful symmetry (which can be described by the alternating group A5 or, equivalently, PSL ( 2, 5) ) , but it also pr ovides a good example of an application of a graph theory model to chemistry. 1 09

7. EIGENVALUES OF SYMMETRICAL GRAPHS

1 10

Before we proceed, we start with some definitions. For a graph G, an automor­ phism f : V (G} � V(G) is a one-to-one mapping which preserves edges , i.e., for u, v E V (G} , we have {u, v} E E if and only if {f(u) , f(v) } E E.

A graph G is vertex-transitive if its automorphism group Aut ( G) acts transi­ tively on the vertex set V (G} , i.e. , for any two vertex u and v there is an automor­ phism f E Au t ( G) such that f(u} = v. We sometimes write f(u} = fu ( especially when both f and u are elements of a group) . We say that G is edge-transitive if, for any two edges {x, y } , {z, w} E E(G} , there is an automorphism f such that {f(x) , J (y) } = {z, w} . It is not difficult to show that an edge-transitive graph is either vertex-transitive or bipartite (or both) .

A homogeneous graph is a graph r together with a group 11. which acts tran­ sitively on its vertex set. In other words, 11. is a subgroup of Aut ( f ) and for any two vertices u, v E V(f} there is a g E 11. such that gu = v. (Since G is the favorite common notation for both graph theorists and group theorists, we will try to avoid using G for either the graph or the group in connection with a homogeneous graph.) Any vertex-transitive graph r can be viewed as a homogeneous graph by taking A ut ( f) as the associated group. However, the choice of the associated group gives different ways of labelling the vertices in the following sense: In a homogeneous graph f with the associated group 11., the isotropy group I is defined as I=

{g E 11. : gv = v}

for a fixed vertex v. We can identify V with the coset space 11./I. The edge set of a homogeneous graph r can be described by an edge generating set K c 11. so that each edge of r is of the form { v, g v} for some v E V and g E K . For undirected graphs we require g E K if and only if g- 1 E K.

Cayley graphs are homogeneous graphs in which the isotropy group I is triv­ ial, I = { id} . In other words, the vertices of a Cayley graph are labelled by its associated group . For example, the cycle Cn can be viewed as a homogeneous graph with associ­ ated group Z and isotropy group nZ . Of course, Cn can also be viewed as a Cayley graph with associated group Zn = Z /nZ. A symmetric graph is regular, so that its Laplacian satisfies

1

£ = I - -A

k

where

k

is the degree and A is the adjacency matrix.

7.2. Cheeger constants of symmetrical graphs In this section , we will discuss several inequalities for Cheeger constants of vertex-transitive and edge-transitive graphs. The various versions of the following inequalities can be found in many places (see Babai and Szegedy [14] } . Here we give a s im ple proof of a strengthened version .

7 . 2 . CHEEGER CONSTANTS OF SYMMETRICAL GRAPHS

111

THEOREM 7 . 1 . Suppose r is a finite edge-transitive graph of diameter D . Then the Cheeger constant hr satisfies 1 hr > ­ 2D '

PROOF. Let S denote a subset of vertices such that lSI $ � where n = I V ( f)l. We consider a random (ordered) pair of vertices (x, y) , uniformly chosen over V ( r) x V ( r ) . Now, we choose randomly a shortest path P between x an d y (uniformly chosen over all possible shortest paths) . Since r is edge-transitive the probability that P goes through a given edge is at most v�fr . -

A path between a vertex from S and a vertex from S must contain an edge in E(S, S ) . Therefore we have 2 I E(S, S ) I · D � Prob(x E S, y E S vol r

-

or

x E S, y E S).

This implies

IE(S, S ) I

>

E(S, S ) k i SI

> >

I SI I B i vol r Dn 2

lS I

Dn 1 . 2D

Therefore

1 hr > - . 2D -

0

THEOREM 7 .2 . Suppose r is a finite vertex-transitive graph of diameter D and degree k. Then the Cheeger constant hr satisfies 1 hr � 2kD . P ROOF. We follow the notation in the previous proof. The automorphism group defines an equivalence relation on the edges of r . Two edges e1 , e 2 are equivalent if and only if there is an automorphism 1r mapping e1 to e2 . We can then consider equivalence classes of edges, denoted by E1 , , E• . ·

We define the index of r to be

index r = mj n

·

·

vol r

2I Ei l

where Ei denotes the i-th equivalence class of edges. Clearly, we have k. In particular, when r is edge-transitive, we have i n de x r = 1.

1 $ index r :::;

Since r is vertex transitive, each equivalence class contains at least i edges . Let Pi denote the probability that a pair of vertices is an edge in the i-th equivalence

1 12

7. EIGENVALUES OF SYMMETRICAL GRAPHS

class E; . Since all edges in the same equivalence class have the same probability, we have, for each i , 1 2 index r . < - < -�-=-­ P vol r E I I • ;

For a subset S of the vertex set with volS � volS, and, for a pair of vertices x, y in r (G) , the probability of having one of x, y in S and the other in S is the same as the probability that P(x, y) contains an edge in E(S, S) . Therefore, we have

Prob(x E S, y E S

or

x E S, y E S)

< <

Since

Prob(x E S, y E S

we have

or

I E(S, S) ID max p; •

IE(S, S) I D

2 ·

�:�e; r.

2ISI IS I x E S, y E S) = 2 n

1 1 IE(S, S) l > __ > vol S - 2D inde x r - 2kD

as claimed.

0

In fact, the above proof gives the following slightly stronger result . THEOREM 7 . 3 . Suppose r is a finite vertex-transitive graph of diameter D .

Then

1 hr > - 2D index r

(7.2)

----­

We remark that in the inequality (7 . 2) the term index r cannot be deleted. There are, for example, Cayley graphs r with h r = k'b for some constant c. EXAMPLE 7 . 4 . We consider the Cayley graph with vertex set Z q x Z 2 and edge generating set { (k, 0) : k E Zq} U { (0, 1 ) } . In other words, the graph r is formed by joining two complete graphs on q vertices by a matching. This is the "dumb-bell" graph and its Cheeger constant satisfies 1 1 2

h=

where degree k =

q

-

q

=

-

kD

<< -

D

is large and diameter D = 2.

7.3. Eigenvalues of symmetrical graphs We here derive lower bounds for eigenvalues of edge-transitive and vertex­ transitive graphs. These bounds are somewhat stronger than the bounds one ob­ tains for general graphs by using Cheeger inequalities . TH EOREM 7 . 5 . For an edge-transitive graph r with diameter D and degree k,

we have

7. 3 .

EIGENVALUES OF SYMMETRICAL GRAPHS

THEOREM 7 . 6 . For a vertex-transitive graph A I 2:::

r

with diameter

113

D,

we have

1

kD2 .

The above two theorems are special cases of the following theorem:

THEOREM 7 . 7. For a vertex transitive graph f with diameter D, we have AI

1

>

D2 in dex r

-=-:::--:--:---= :

-

where in dex r = min �fk5 , and where E; denotes the i-th equivalence class of edges under Aut ( f ) . P ROOF . We consider f : V (r) definition ( 1 .5) in Chapter 1 :

AI

IR and we use the (equivalent) eigenvalue

-t

n L )f(x ) - /(y)) 2 = min 11 1 k L L ( f (x ) - / ( y)) 2

"' -=------

---

z

For each edge

II

e = { x, y }, we define f (e ) = if (x ) - f( y) i.

We then have

, = min "I ·

I

n L ! 2 (e ) eEE

k L L ( f (x ) - f(y)) 2 z

.

II

Let E; denote the i-th equivalence class of edges under Aut ( f ) . For a fixed vertex

xo, we choose a fixed set of shortest paths P,0,11 to all y in r. We can now use the automorphism group to define, for each vertex x E V(r ) and an automorphism 1r with rr(x0 ) x, set of paths P(x ) = {rr(Pzo,ll ) }. Clearly, each path i n P(x ) h as length at most D. For each edge e, we consider the number N. of occurrences of e in paths in P(x ) ranging over all x. Two edges in the same equivalence class have the same value for N. . The total number of edges in all paths in P(x) for all x is at most n2 D. For each i and e E E; , =

a

we have

N.

n2 D

n2 D

lEd - 2IE; I - 2 min ' <

--

<

-

nD index r

r

-----

vol

Now we consider, for a harmonic eigenfunction

D index r k

f achieving >. I ,

7. EIGENVALUES OF SYM METRICAL GRAPHS

1 14

"'

z

!I

eEP(z, y )

!I

L L D L.: z

<

P (e )

eEP(:�: ,y)

!I

L,: P (e ) DNe

<

eE E

L f2 ( e ) D

�

D inde x r ·

eEE

k

Therefore we have k L L,: U (x) z

-

f (y ) ) 2

!I

1 > 2 D index r · This completes the proof of Theorem 7. 7.

0

7.4. Distance transitive graphs We recall that a graph r is distance transitive if for any two pairs of vertices {x , y } , { z, w } with d(x, y) = d(z, w) , there is a graph automorphism f satisfying f(x) = z and f(y) = w. In this section, all graphs we consider are distance transi­ tive. We first point out an easy observation which has many useful consequences.

Observation: Let A denote an eigenvalue of a distance transitive graph r . Suppose we fix a vertex v in f. Let V; = {u E f : d(u, v) = i} Then A is achieved by an eigenfunction which satisfies for each i , x, y E V; implies f(x) = f(y) . This observation follows from the fact that for any x, y E V; there is an automor­ phism mapping x to y that fixes v. If we start from any eigenfunction f achieving A , the new function formed by assigning the average value in V; to vertices in V; is also an eigenfunction with the same eigenvalue.

THEOREM 7 . 8 . The eigenvalues of a distance transitive graph f of diameter D are the eigenvalues of a weighted path P of D + 1 vertices. P ROO F . We form a path P by contracting each set V; as defined above into one vertex . The weight of the edge { v; , vi+ 1 } in P is just the number of edges between V; and V;+l . The loop { v; , v; } has weight equal to twice the number of edges in r with both endpoints in V; plus the number of loops in V; . For an eigenfunction f of r with eigenvalue A, we choose v so that f (v) is nonzero and we map v into v0 of the path P. Then we assign the average value of f ( over the vertices in V; ) to v; in P. Clearly, the resulting function is not always 0 and is a harmonic eigenfunction 0 of P with the same eigenvalue .X.

7.4.

DISTANCE TRANSITIVE GRAPHS

115

It is, of course, much easier t o deal with a ( D + 1 ) x ( D + 1 ) matrix of the path P with nonzero entries only at coordinates of (i, i + 1 ) , (i, i), and (i, i - 1) , than to deal with the Laplacian of G. An eigenfunction for P can be transformed into an eigenfunction for r as follows:

THEOREM 7. 9 . A harmonic eigenfunction for the contracted path P is an eigen­ function for the distance transitive graph r . P ROOF. Suppose f i s a harmonic eigenfunction for P. We define if X is in V; . It is easy to check that g is an eigenfunction for f.

g( x) = f (v; )

0

As a consequence of Theorem 7.8 and 7.9, we have

THEOREM 7 . 1 0 . A

distinct eigenvalues.

distance transitive graph of diameter D has exactly D + 1

P ROOF . From Theorem 7.9, we know that there are at most D + 1 distinct eigenvalues. On the other hand, there are at least D + 1 distinct eigenvalues because otherwise the characteristic polynomial IT (£ - .X;) would be of degree :5 d, contradicting the fact that

I, £, £2,

>.; distinct

• • •

, £ d are independent .

0

In fact , a slightly stronger statement is true.

THEOREM 7 . 1 1 . The eigenvalues of a distance transitive graph f of diameter have the same values as the eigenvalues of a weighted path P of D + 1 vertices, obtained by contracting each V; = { u d( u, v ) = i}, for a fixed v, into a single vertex.

D

:

Of course, the eigenvalues of r and P have different multiplicities. Nevertheless, there is a simple method to determine the multiplicities of ei genvalues of r from eigenfunctions of P. Note that in P, all eigenvalues have mult iplic ity 1 .

THEOREM 7. 1 2 . In a distance transitive graph r, we consider the n n matrix with A1 (x, y ) = 1 if d(x , y) j in r and 0 otherwise. The eigenvalues for A1 are I Vi l f(vj ) / f (vo) for each harmonic eigenfunction f of P . x

Aj

=

PROOF . Since A1 is a polynomial in A, then an eigenfunction f of A is an eigenfunction for A1 . From Theorem 8.9, a harmonic eigenfunction f of P is an eigenfunction of A, i.e . , f ( x) = f ( v ) for x E Vv . Suppose the eigenvalue of f for Ai is A' . We consider, for the vector t.po = (1 , 0, . . . , 0 ) of length D + 1 , the inner product (¢0 , 1rAj /) where

1r

=

(¢o , A17r/) = J.. ' f ( vo )

denotes the projection of V(f) into P, i.e. , 1rj(v;) =

L f (x ) .

x E V;

116

7 . EIGENVALUES OF SYMMETRICAL GRAPHS

We remark that 1r can be viewed as a matrix with rows indexed by V (P) and columns indexed by V{f) such that 1r{ v , x ) = 1 if x E V., and 0 otherwise. Clearly, we have

(ifJo7rAj , J) = I Vi l f(vJ ) · Therefore, >.. ' = I Vi l f(vi ) / f ( vo ) .

D

Using Theorems 7.9, 7.10 and 7.12, we will show that the multiplicities of each eigenvalue can be determined from the contracted path P.

THEOREM 7 . 1 3 . For a distance-transitive graph r on n vertices, the multiplic­ ity m( >.) of an eigenvalue >.. is m ( >.. ) = II 11 2 l ( vo) where g is the corresponding eigenfunction of the weighted path P contracted from f from a fixed vertex

;

Vo .

PROOF . Let g/s denote eigenfunctions of P and J;'s denote the corresponding harmonic eigenfunctions. Note that g( vi ) = JkiVJf(vi ) where k is the degree of r. We consider the following n x n matrix: D

Mi = L fi (vj )Ai . j =O For a fixed i, the trace of Mi is just the trace of fi(v0 ) A0. Hence tr ( Mi ) = nfi(vo ) . ( 7 . 3) On the other hand, the eigenvalues of Ai are I Vi lfo ( vi ) / fo (vo), I VJ I II ( vi ) / II ( vo ), . . . , I Vi l fv ( vj ) / fv( vo ) of respective multiplicities m ( >..0 ) , , m ( >.. v ) . Therefore, . . •

tr M

D

=

L fi(vj ) tr AJ

J =O

� f; (vJ ) � m (>.t ) fl (vj ) fll V(�o)I D

=

D

D

I· Vi l L fi( vJ )m ( >.; )fi( vj ) f , ( Vo ) J= O ( >.. ; ) l l g; ll 2 /im( vo) k "

Here, we use the fact that distinct eigenfunctions are orthogonal so that

"" /p (vi ) /q(vi)I Vi l k = { � •

Since

0

I gp l l 2

g2 ( vo) = k f 2 (vo) , from (7.3) , we have m ( >.. ) = I I 11 2 g2 ( vo ) .

The proof of Theorem 7.13 is complete.

;

if p # q if p = q .

D

7.5.

EXAMPLE

1 17

EIGENVALUES AND GROUP RE P RESENTATION THEORY

7. 1 4 .

The Petersen graph and intersection gmphs

The Petersen graph is a distance transitive graph on 10 vertices . The vertex set is labelled by all 2-subsets of { 1, 2, 3, 4, 5}. Two vertices A and B are adjacent if and only if A n B = 0. The eigenvalues are 0, i (with multiplicity 5 ) , and � (with multiplicity 4) . The Petersen graph is a special case of the intersection graph G (n, r, k) with vertex set consisting of all r-subsets of { 1 , . . . , n } . The vertices A and B are adjacent if I A n B I = k. The symmetric group Sn acts on the graph with isotropy group Sr X Sn-r· Since (Sn , Sr x Sn-r) is a Gelfand pair [97] , the spectral decompositions of the space :F(v) = {! : V -t R} are quite special: :F(v) = Eo

EB

E1

where the dimension of Ei satisfies dim Ei

. EB Er (7) - {;�\) for i � 1 , and dim E0

EB

=

.

.

=

1.

7 . 5 . Eigenvalues and group representation theory A brute force method for computing eigenvalues of a connected graph on n vertices is to solve for x in the determinant, det (x l - .C), of an n x n matrix. Before starting such an arduous task, it makes sense to see if the matrix can be diagonalized into smaller blocks. Group representation theory is exactly the answer to such prayers when the graph is homogeneous. Suppose r is a Cayley graph (25]with vertices labelled by a group 1-£ and with edge generating set K. Let p denote an irreducible representation of 1-£ of dimension l . This means that p maps the elements of 1-£ into l x l matrices in such a way that matrix multiplication is consistent with the group multiplication, i.e., p(g 1 g2 ) = p(gt ) p (g2 ) . The eigenvalues of r are exactly the eigenvalues of the smaller matrix I-

1

TkT L p (g)

p ranging over all irreducible representations of 11. . Each dim p di p matrix has multiplicity dim p in the graph r. g E K.

for

X

eigenvalue of the

m

Suppose r is a homogeneous graph with associated group 1-£ . The vertex set can be identified by 1-£/I where I is the isotropy group. The edge generating set K = {gi : v gv} for a fixed v is a union of double cosets .....,

K = IKI T he eigenvalues of r are the eigenvalues of I degree of r.

-

l �l L p( x ) where � gL x Egi iE K.

k is the

The best way to illustrate the connection between homogeneous graph r and the irreducible representations of the associated group 11. is by examining concrete examples:

EXAMPLE 7 . 1 5 .

The cycle Cn as a Cayley graph associated with Lapl acian can be diagonalized since the irreducible representations PA: (g) = (e 2 :•k , g), k = 0, . . , n - 1 , are all !-dimensional. .

Z n.

The

7 . EIGENVALUES O F SYMMETRICAL GRAPHS

118

We remark that for the Gelfand pairs in Example 7.15, all irreducible repre­ sentations are !-dimensional. This simplifies the computation of the eigenvalues of the corresponding homogeneous graphs.

..

In our final example we use terminology that may be unfamiliar to some readers.

A quick summary of this can be found in (85] or (84, 86] .

EXAMPLE 7 1 6 The Buckyball, a soccer ball-like molecule, consists of 60 car­ bon atoms . It corresponds to a Cayley graph on A5 with edge generating set { ( 12345) , (5432 1 ) , ( 1 2) (23) } . The edges generated by ( 12) (34) correspond to "dou­ ble bonds" and the edges generated by (12345) , (54321 ) to "single bonds" . The irreducible representations for the alternating group were determined by Frobenius [134] and they are of dimensions 1, 3, 3, 4, and 5. This means the Laplacian can be diagonalized into blocks of sizes 1 x 1 , 3 x 3 (with multiplicity 3 ) , 3' x 3' (a second type with multiplicity 3) , 4 x 4 (with multiplicity 4), and 5 x 5 (with multiplicity 5) . Note that 1 2 + 3 2 + 3 2 + 4 2 + 5 2 = 60 .

Suppose we consider the weighted graph with single bonds of weight 1 and double bonds of weight t . The eigenvalues of the adjacency matrix are exactly the eigenvalues of pa + pa- 1 + tpb for any irreducible representation p and a = (12345) , b for the dimension 5 representation p5 we have

p, (a) �

�

(12) (34) . For example,

0 � � � �)

(�

-1

p, (b)

=

-1

0 0 0 1

-1 0 1 0 0

-1 0 0 1 0 1-t 0 1 0

-1 1

0 0 0

-t 1 t 1 0

) -

0 1 t 1

t

T)

7.6. THE VIBRATIONAL SPECTRU M OF A GRAPH

119

Thus, the eigenvalues of the adjacency matrix are the roots of the characteristic polynomial

(x2 + x - t2 + t - 1)(x3 - tx2 - x2 - t2x + 2tx - 3x + t3 - t2 + t + 2) . In summary, the eigenvalues of the Buckyball can be written in closed form as roots of the following equations where the single bonds are weighted by 1 and the double bonds are weighted by t:

(x2 + x - t2 + t - 1) (x3 - tx2 - x2 - t2x + 2tx - 3x + t3 - t2 + t + 2) = 0

(a) :

with multiplicity 5;

(x2 + x - t2 - 1) (x2 + x - (t + 1)2) = 0 with multiplicity 4; (x2 + (2t + 1)x + t2 + t - 1)(x4 - 3x3 + (-2t2 + t - 1)x2 + (3t2 - 4t + 8)x + t4 - t3 + t2 + 4t - 4 = 0 with multiplicity 3; (d) : x - t - 2 = 0 with multiplicity 1. (b) : (c ) :

76 .

.

The vibrational spectrum o f a graph

The Laplacian £ of a graph r is an operator acting on the space of functions -+ IR} . A natural generalization is the vibrational Laplacian Cx which acts on the space .F (V, X ) = { f : V ( f ) -+ X } for some vector space X . We use the word "vibrational" since the spectrum of the vibrational Laplacian £x of a graph r for the special case of X = IR3 is exactly the vibrational spectrum of the molecule whose atoms correspond to the vertices of r and whose bonds between atoms are just edges of r [84).

{! : V(f )

We start with a homogeneous graph r with the associated group H and isotropy group I. We can generalize the Dirichlet sum (see Section 1.2) as follows : Suppose that to each edge, e = { u, v}, of the graph we associate a self-adjoint operator , A. , on X ; we then define the quadratic form Q on .F(V, X) by

(g

(7. 4)

, £x g =

)

1

2 � )g (u) - g(v)] · A. · [g(u) - g (v)]

where the sum ranges over all edges of r. Suppose on X . Furthermore , suppose Ae in (7.4) satisfies

p denotes a representation of H

Aae = p(a)A.p(a)-1 where ae denotes the edge { ab, ac} and the edge e is denoted by { b , c}. Then i t can be shown (see [84)) that the spectrum of Cx can be decomposed into the union of the spectra of the following operators over all irreducible representations

where

( ) L Ag g EIC

(7.5) 0

0 I -

1

of

r:

L A g p (g ) 0 1(9) gEIC

denotes the cartesian product (of matrices) .

Now we return to the example of the Buckyball graph. We will apply Hooke's law to derive the vibrational spectrum of this graph. Let u E IR3 and w E IR3 denote the equilibrium positions of the VP.rtices labeled u and w . Let h : \' -+ IR3

1 20

7. EIGENVALUES OF SYMMETRICAL GRAPHS

describe a deviation from equilibrium, so that u + h(u) is the new position of the vertex u . Then the potential energy associated to h can be expressed as: 1 W(h) = L ku , w ( l lu + h(u) - ( w + h (w) ) ll - l l u - w ll ) 2 .

2

In the above expression, the sum is over all pairs { u, w} of vertices connected by an edge, and ku, w is the spring constant of that edge. IT h is sufficiently small so as to enable us to ignore terms quadratic in h, we then have ll u + h(u) - (w + h(w) ) ll � (l l u - wll 2 + 2(u - w) ( h( u) - h (w) )] � ·

l l u - w l l + Wu,w (h (u ) - h(w) )

�

·

where

u-w ll u - w l l is the unit vector from u to w and denotes the scalar product on IR.3 • Then the quadratic approximation to W is given by 1 W ( h) = ku,w [Wu ,w ( h(u) - h(w) W . 2 Hence, we may take A e to b e a 3 x 3 matrix: Wu,w = ·

L

Ae = Wu, w

where

e

is the edge joining u to

·

18)

w�.w

w.

We can now use the above methods and {7.5 ) to compute explicitly the vibra­ tional spectra of a molecule in terms of the irreducible representations of A5 . The space of displacements is .F(V, IR.3 ) = {/ : V --t IR.3 } . We choose p to be the ordinary three-dimensional representation (which is just rotation in IR.3 ) . Using irreducible representations of A5 , we can then evaluate explicitly all vibrational eigenvalues by treating 3 x 3, 9 x 9, 9 x 9, 12 x 1 2 , and 1 5 x 1 5 matrices. We point out that the above methods not only determine the vibrational spec­ trum, but also the specific representation associated to each eigenvalue. This addi­ tional information is important in chemical applications. For the case of homoge­ neous molecules, we can, in advance of all computations and independent of specific models for the potential energy, determine the number of representations of each type by a simple application of the Frobenius reciprocity formula. Now the space of displacements of the Buckyball has dimension 180 = 60 x 3. B u t the space o f entire (infinitesimal ) rigid displacements o f the molecule as a whole is six-dimensional (the Lie algebra of the Euclidean group ) . By subtracting these six dimensions, we get

1. The space of vibrational states is 1 74-dimensional. The 180-dimensional space is the tensor product of the regular representation with a three-dimensional representation . So we must decompose the regular repre­ sentation , which contains each irreducible representation with a multiplicity equal to its dimension . We have already mentioned the irreducibles of A 5 which we here denote as 1 , 3, 3',4 and 5, where 3 is the three dimensional representation given by the action of A5 on the icosahedron, and 3' differs from 3 in that the generator of degree 5 has been replaced by its square. Since 1 + 3 + 3 + 4 + 5 = 16, we see that

7 . 6 . THE VIBRATI O N A L S PECTRUM OF A G RA P H

121

the 180-dimensional displacement space decomposes into 48 = 3 x 16 irreducibles. Subtracting off two three-dimensional representations, we obtain: 2.

The number of distinct vibrational modes is at most ,46.

For a vibrational line to be visible as an absorption or emission line in the infrared (as a transition between the ground state and a one-photon state) it is necessary that the associated irreducible representation be equivalent to (the com­ plexification of) the representation of 1i on the ordinary three-dimensional space JR3 in which the molecule lies . In the Raman experiment, light of a definite frequency is scattered with a change of frequency. This change, known as the Raman spectrum, is associated to those representations which intertwine with the space 82 (JR3 ) of symmetric two tensors. Therefore, both the infrared spectrum and Raman spectrum can be directly determined by using the Frobenius reciprocity formula. For details on this , the reader is referred to [84, 230] . 3. The space of classical vibrational states has dimension 1 7,4. Any force matrix, F, invariant under the group A5 has {at most} 46 distinct eigenvalues yielding four lines visible in the infrared and ten in the Raman spectrum.

Notes: The proofs in Section 2 are mainly adapted from [14] . There are several chap­ ters on distance transitive graphs in Biggs [25] . Here we have given slightly different proofs. The computation for the spectrum of the Buckyball graph is based on [87] . More reference on the vibrational spectrum of graphs can be found in [84, 85 , 86] .

CHAPTER 8

Eigenvalues of subgraphs with boundary conditions

8 . 1 . Neumann eigenvalues and Dirichlet eigenvalues In a graph G, for a subset S of the vertex set V = V(G) , the induced subgraph determined by S has edge set consisting of all edges of G with both endpoints in S. Although an induced subgraph can also be viewed as a graph in its own right , it is natural to consider an induced subgraph S as having a boundary. (Here and throughout, we shall denote by S the induced subgraph determined by S, when there is no danger of confusion. ) There are two types of boundaries. The vertex boundary 8S of an induced subgraph S consists of all vertices that are not in S but adjacent to some vertex in S (also defined in Chapter 2 ) . The edge boundary, denoted by as' consists of all edges containing one endpoint in s and the other endpoint not in S, but in the "host" graph. The host graph can be regarded as a special case of a graph with no vertex boundary and no edge boundary. For an induced subgraph S with non-empty boundary, there are, in general, two kinds of eigenvalues - the Neumann eigenvalues and the Dirichlet eigenvalues , subject to different boundary conditions .

Neumann boundary condition: The Laplacian C. acts on functions f : S u oS --+ IR with the Neumann boundary condition, i.e . , for every x E OS

L (J( x ) - f ( y )) = 0 .

(8.1)

y E S,y�x

Dirichlet boundary condition:

The Laplacian [. acts on functions with the Dirichlet boundary condition. In other words, we consider the space of functions { ! : V --+ IR} which satisfy the Dirichlet condition f (x)

(8.2) for any vertex

x

=

0

in the vertex boundary 8S of S.

We remark that the Dirichlet boundary condition for graphs corresponds natu­ rally with that for smooth manifolds. In fact, the general condition with boundary value specified by a given function on 8S will be discussed in the last section of this 123

1 24

8. EIGENVALUES OF SUBGRAPHS WITH BOUNDARY CONDITIONS

chapter. As we will see, problems with the general boundary condition can often be reduced to problems with the Dirichlet boundary condition (8.2) . The Neumann boundary condition (8. 1 ) corresponds to the Neumann boundary condition for Riemannian manifolds: 8v

8f(x)

=

0

for x on the boundary where v is the normal direction orthogonal to the tangent hyperplane at x . Neumann eigenvalues are closely associated with random walk problems whereas the Dirichlet eigenvalues are related with many boundary-value problems. 8.2. The Neumann eigenvalues of a subgraph Let S denote a subset of the vertex set V (G) of G. Let S* denote the union of the edges in S and the edges in 8 8. We define the Neumann eigenvalue of an induced subgraph S as follows:

L (f(x) - f(y)) 2

=

(8.3)

inf N- 0 E . e s /( z)d. = O

{z,y}ES•

x ES

L ( f (x) - f(y) ) 2

.mf sup ...;.::-'-:= ..._ -----N- 0 c L (f(x) - c) 2 dx xES I n general, w e define the i-th Neumann eigenvalue As,; t o be { z ,y}ES•

L (f(x ) - f(y ))2

{ x ,y}ES• , ;= m . f sup -'=:::'-------"S o ,

N- f' E C, _ l

L (f(x) - f '( x)) 2 d,

:rES

where ck is the subspace spanned by functions .s,j , for 0 � j � k . Clearly, >.s, o = 0. We use the notation that >.s, 1 = As and we remark that d:r still means the degree of x in G (independent of S) .

From the discrete point of view, it is often useful to express the ..\s, ; as eigen­ values of a matrix .C s . To achieve this, we first derive the following facts: LEMMA 8 1 Let f denote a function f eigenvalue .A . Then f satisfies: .

(a) :

.

for x E S, L f (x )

=

:

S U §S

-t

R

satisfying

L (f(x) - f (y ) ) = >.f ( x ) dx ,

y {x,y}E s ·

(8.3)

with

8.2. THE N EUMANN EIGENVALUES OF A SUBGRAPH

(b) :

for x E 6S,

1 25

L (f(x) - f ( y)) = 0 . II

{z,y}E8S

This is the Neumann condition. Equivalently, f(x) = d1' L f(y) X

If

{ :r , y } E 8S

where d� denotes the number of neighbors of x in S. for any function h S U 6 S -+ Ill, we have L h(x)L f (x) = L (h (x) - h (y) ) (f (x) - f(y) ) .

(c) :

:

·

xES

{ x ,y } E S •

We remark that the proofs of (a) and (b) follow by variational principles and that (c) is a consequence of (b) . Using Lemma 8.1 and equation (8.3) , we can rewrite (8.3) by considering the operator acting on the space of functions {! : S -+ Ill} , or, alternatively, on the space of functions {! : S U 6S -+ Ill an d f satisfies the Neumann condition} :

L f (x ) Lf (x )

{8. 4)

>.s

_:rE"":=::S ---

inf

"2'. /(� d.=O :rL f 2 ( x)dx ES L g(x)£g( x) x ES inf

=

g .LTI/ 2 1

=

inf 9.tT' 12 1

( g , [.g} s (g, g}s

where £ is the Laplacian for the host graph G and (It , h}s

L It (x)h (x) .

=

zES

For X c V , we let Lx denote the submatrix of L restricted to the columns and rows indexed by vertices in X. We define the following matrix N with rows indexed by vertices in S U 6 S and columns indexed by vertices in S.

N (x , y )

=

{�

Further, we define an l SI

1 d"'' 0

x

if X = y if X E S and

X

::/:

y

if x E 6S, y E S an d otherwise

lS I matrix

2 2 •r JV S = y- t f N* L SU6S Nr - t /

x

,....,

y

126

8 . EIGENVALUES OF SUBGRAPHS WITH BOUNDARY CONDITIONS

where N• denotes the transpose of N . It is easy to see from equation (8.4) that the �S,i are exactly the Neumann eigenvalues of Ns . In fact Ns has an eigenvalue 0 and, if S is connected, it has l SI - 1 positive eigenvalues.

8.3. Neumann eigenvalues and random walks For an induced subgraph S of a graph G, we consider the following so-called Neumann random walk: The probability of moving from a vertex v in S to a neighbor u of v is 1/ dv if u is in S. If u is not in S, we then move from v to each neighbor of u in S with the ( additional ) probability 1 /dvd� where d� denotes the number of neighbors of u in S. The transition matrix P for this walk, whose columns and rows are indexed by S, is defined as follows:

(8.5 )

The stationary distribution is dv (L. du at the vertex v . We remark that above u Neumann walk is somewhat different from the random walks often used ( in which if v has r neighbors not in S, then the probability of staying at v is r fdv . ) In a way, the Neumann walk takes advantage of "reflecting" from the boundary as dictated by the Neumann boundary condition. The eigenvalues Pi of the transition probability matrix P associated with the Neumann walk are related to the Neumann eigenvalues �S,i as follows:

Pi =

1 - �S,i·

In particular , we have

( 8 .6)

p

= P1 = 1 - �S, l

=

1 - �S·

8 . 3 . NEUMANN EIGENVALUES AND RANDOM WALKS

This can be proved by using the Neumann condition

1-p

as

127

follows :

L (f(x) - J(y)) 2 + L (f(x) - J(y)) 2 /dz z-z- y :r�y :r,yES :r,yES,z(/.S inf / fO L ! 2 ( x)d:r zES E u< x) - J (y )) 2 + E L [d:J 2 (x) <E J (y)) 2 lf d� :r�y :r,yES inf -

>

#0

:rES

>

:rES L (f (x ) - f(y)) 2 + L L U(x ) - f(z)) 2 >

:rES

>.. s

:rES

where f ranges over all functions f : 6S U S -+ lR satisfying =

L f( x )d:r

=

0,

zES and for

x

E 6S

L ( J(x ) - f( y ) )

=

0.

yE S,y-:r

Inequality (8.6) is quite useful in bounding the rate of convergence of Markov chains for problems which can be formulated as random walks in a subgraph with a bound­ ary. Suppose S is an induced subgraph of a k-regular graph . The above random walk can be described as follows: At an interior vertex v of S, the probability of moving to each neighbor is equal to 1/k. (An interior vertex of S is a vertex adjacent to no vertex outside of S . ) At a boundary vertex v E 6S , the probability of moving to a neighbor u of v is 1 / k if u is in S, and , in this case where u ¢ S, an (additional) probability of 1 / ( k d� ) is assigned for moving from v to each neighbor of u in S. The stationary distribution of the above random walk is just the uniform distribution .

8 . EIG ENVALUES OF SUBGRAPHS WITH BOUNDARY CONDITIONS

128

8.4. Dirichlet eigenvalues We consider a graph G with vertex set V = V (G) and edge set E = E (G) . Let S denote a subset of V and we assume that the vertex boundary 6S is nonempty. We use the notation condition

I E n·

to denote that

l( x )

=

I satisfies the Dirichlet boundary

0 for x E 6S.

The Dirichlet eigenvalues of an induced subgraph on S are defined as follows:

L ( f ( x ) - l(y)) 2 z { ,y}ES" inf //�· L 12( x )dx :tES

(8.7)

inf g-#0

gE D"

. mf g-#0

=

gED"

(g , £g) (g, g)

:tES

In general, we define the i-th Dirichlet eigenvalue A; to be - ( D)

L ( f (x) - I ( Y)?

{·'" ·Y }_E_ 1n f sup -:: ::::::,::: S_•-----

' = 1"1-0/' EC;-t L (/( x ) - f '(y)) 2 dz z ES

A·

•

where C; is the subspace spanned by eigenfunctions tPi achieving Aj , for 1 � j � i. We use the notation ,\ �D) = ,\': ) . We will index the Dirichlet eigenvalues for a subgraph on l S I vertices by 1 � i � l S I . With very similar proofs to those in Chapter 1 , i t can b e shown that for a connected induced subgraph S of a graph G with a s -I- 0, we have 0 <

SI < D A ( ) < I8 1 1

-

vol S -

0 < A .(D)

for 1 � i � lS I .

<

-2

LEMMA 8 . 2 . For an induced subgraph S, let g denote an eigenfunction of £ with Dirichlet eigenvalue .\, i.e., g S --+ lR satisfies (8.7) and the Dirichlet boundary con dit i on g ( x ) = 0 for x E 6S . Then g satisfies :

8 . 5 . A MATRIX-TREE THEOREM AND DIRICHLET EIGENVALUES

( 1 ) : for x E 8,

1 Cg(x ) = ­

129

(x) g( ) - y ) (g Vd; ...jd;

L

= Ag(x) y s {:z:,y}e • (2) : for any function h : V --+ IR, h h L h (x)Cg(x) = L ( (x ) - (y ) ) . ( g(x) - g( y ) ) :z: ES Vii; .._fil; { :z: ,y } E S• Vii; .._fil;

Vii;

The proof of (I) follows from the variational principles and will be omitted. To see (2) , we note that

L h (x ) Cg(x ) - L

:z:

Es

L :z: E.SS

0.

(

{ :z: .11} e s• g (x) h ( x) _ h( y) ( )

Vii; Vii;

h (x) - h (y) . g (x) - g (y) ) ( )

Vii;

...jJ;

Vii;

...jJ;

.._fil;

A s ari immediate consequence of Lemma 8.2, for functions f the assumption that f(y) = 0 for y E 88, we have, for all x E S,

:

8

--+

IR under

C f( x ) = Csf(x) where Cs is the submatrix of C restricted to columns and rows indexed by vertices in S. We note that since 88 =I= 0, Cs is nonsingular. All eigenvalues of Cs are positive. Hence The Dirichlet eigenvalues of the induced subgraph on S are just the eigenvalues of Cs arid the determinarit of Cs can be expressed as : l SI

det Cs

=

IT ,\;n) . i=l

8-5- A matrix-tree theorem and Dirichlet eigenvalues One of the classical combinatorial theorems is the matrix-tree theorem which states that the determinant of any cofactor of the combinatorial Laplaciari is equal to the number of spar�ning trees in a graph. This result Carl be traced back to Kirchhoff [171] ; see also Maxwell [196] , Sylvester (232] arid Cayley (47] . There is a large literature of related work such as Tutte (239] , Bott and Mayberry (39] , Trent [238] , Uhlenbeck arid Ford (240] , Renyi [218], and Moon (200] . In this section, we consider a generalization of the matrix-tree theorem for induced subgraphs of a graph ( see (89]) . For an induced subgraph S with non­ empty boundary in a graph G, we define a rooted spanning forest of 8 to be any subgraph F satisfying: ( 1) F is ari acyclic subgraph of G, (2) F has vertex set 8 U 88, (3) Each connected component of F contains exactly one vertex in 88.

8.

130

EIGENVALUES OF SUBGRAPHS WITH BOUNDARY CONDITIONS

The following theorem (see (89) ) relates the product of the Dirichlet eigenvalues of S to the enumeration of rooted spanning forests of S : THEOREM 8 . 3 . For an induced subgraph number of rooted spanning forests of S is

II d" xE S

S

in a graph

G

with

§S =f. 0,

the

lSI

II .x;

i=l

where .\; , 1 � i � l S I , are the Dirichlet eigenvalues of the Laplacian of S in G . P RO O F . We consider the incidence matrix B with rows indexed by vertices i n

s and columns indexed by edges in s· defined as follows : B(x, e) =

We have

{�

e

=

1 __

if e

=

0

otherwise.

v'd;

£

(8.8)

if

{x, y } , x < y {x, y } , x > y

B B*

=

where B* denotes the transpose of B . Then

lSI

II .\ ;

i=l

det £ det B B * L det Bx det B)c X

where X ranges over all possible choices of s - 1 edges and Bx denotes the square submatrix of B whose s 1 columns correspond to the edges in X . -

Claim 1: I f the subgraph with vertex set S U § S and edge set X contains a cycle , then det Bx 0. The proof follows from the fact that the columns restricted t o those indexed b y the cycle are dependent. =

Claim !2: If the subgraph formed by edge set X contains a connected component having two vertices in §S, then det B x = 0. Proof: Let Y denote a connected component of the subgraph formed by X . If Y contains more than one vertex in §S, then Y has no more than IE(Y) I 1 vertices i n S. The s u b m a tr i x formed by the columns corresponding to edges in Y has rank at most IE(} ' ) I 1. Consequently, det Bx 0. -

Claim

-

3:

=

I f the subgraph formed b y

X

I det Bx l

i s a rooted forest o f S , then 1

=

dx ' nxE S v'd;

8 . 6 . DETERM I N A NTS AND I N VA R I A N T F I E L D THEORY

131

Proof: From Claims 1 and 2, we know that edges of X form a forest and each connected component contains exactly one vertex in tSS. There is a column indexed by an edge with only one nonzero entry, say (x1 , e 1 ) with x1 E S. Therefore, l det B ., l =

1

vr::cdx1

1 det B� 11 l l

where B�� ) denotes the submatrix with rows indexed by S - { x i } and columns indexed by X - { e } . By removing one edge and one vertex at a time, we eventually obtain 1 l det B., I = IT ..;;i;. l SI

xES

Combining Claims 1-3, we have

II A ;

i= 1

=

=

de t £

IT

=

L det B

xES x

1

d

x

X

de t Ex

.

! { rooted spanmng forests of S} l .

This completes the proof of Theorem

8.3.

0

We remark that the usual matrix-tree theorem can be viewed as a special case of Theorem 8.3. Namely, for a graph G, we apply Theorem 8 . 3 to an induced subgraph H on V (G) - { v } for some vertex v in G. The rooted spanning forests correspond in an one-to-one fashion to all trees in G .

8.6.

Det erminants and invariant field t heory

Conformal invariant theory for free bosons intimately uses the determinant of the Laplacian with Dirichlet boundary conditions (see [142] , [ 1 76] ) . The discrete version can be described as follows: For an induced subgraph 5 with no n - em pty boundary 85, suppose a is a function defined on the b ou n dar y 85. (Often the function a satisfies some periodic conditions as well . ) The "energy" for a function f is related to

H(f)

=

x�y xES

L [f (x ) - f(y ) ] 2

where the summation ranges over all edges { x , y } a t least in 5. The partition function can be expressed as

Z(a)

=

I e-c

one

endpoint o f which is

H( f l

f ranges over all functions whose restriction on 85 is a . To compute Z (a) , we note that the function fo that minimizes H ( f) an d shares

where

the same value as X E S:

a

on the boundary, also satisfies the following condition for all

l: Uo (x ) - fo(y)) = 0 .

y y� x

13 2

8. EIGENVALUES OF SUBGRAPHS WITH BOUNDARY CONDITIONS

This can be proved by variational principles. Also, it is not difficult to show that such a function exists if S is connected and, when it exists, is uniquely determined. For any function g whose restriction to fJS is a, we consider

I = g - lo

H (g ) as follows: L (/o(x) - lo(y))2

Clearly, I satisfies the Dirichlet condition. We can rewrite

H ( g)

L (f(x) - l(y))2 +

=

{ z ,y } E S•

{ z , y } E S•

L (f(x) - l(y))(lo(x) - lo(y))

+

{z ,y} E S•

L ( f (x) - l(y))2 + L (/o(x) - lo(y))2

=

{ z ,y } E S •

+

L lo (x ) L ( f (x) - l(y))

zeS•

=

{ z ,y} ES•

yES z�y

L (f( x) - l(y)? + L (/o( x ) - lo(y))2

{ z ,y} E S•

{ z ,y} E S •

Therefore, we have z

=

e - c H ( /o )

i e-c

H(f)

where I r anges over all functions s at isfy ing the Dirichlet condition. If we ex press I rel at i ve to the basis formed by orthonormal e igenfuncti ons c/J; of the induced graph S with n vert i ces , we obtain

Thu s ,

H (f) = L: xi A ;

and z

e-c

H (/o )

J e-c E >..; z� dx;

l SI c - n /2 (IJ A; ) -1/2 11" - n/2 . i= 1

Therefore the present problem is reduced to the problem of evaluating the prod ­ of the Dirichlet eigenvalues which, in turn, is the same as enumerating uct IJ\!11 rooted spanning forests of an induced s ubgraph .

A;

The problem of evaluating the determinants of Lapl ac i ans on Riemann surfaces has been considered by many researchers (e.g. , Polyakov [2 1 2] in s t ring theory, Ray-Singer [2 16] in an alyti c torsion, D'Hoker and P hong [95] i n connection with Se lber g ' s zeta fu nct ion , Sarnak (220] , Osgood , Phillips and Sarnak (203, 204] for is ospectral sets ) . The discrete parallel involves a combinatorial trace formula and zeta functions , which however will not be incl uded in th i s book becaus e of space

8.6. DETERMINANTS AND INVARIANT FIELD THEORY

1 33

limitations. The heat kernel of a graph plays a major role there and will be discussed in the next chapter. Notes:

The Neumann eigenvalues and Neumann walks are mainly based on (60] and (79]. More discussion on Dirichlet eigenvalues can be found in (176] and also in (8 9] .

CHAPTER 9

Harnack inequalit ies

9. 1 . Eigenfunctions A crucial part of spectral graph theory concerns understanding the behavior of eigenfunctions. Intuitively, an eigenfunction maps the vertices of a graph to the real line in such a way that edges serve as "elastic bands" with the effect of pulling adjacent vertices closer together. To be specific, let f denote a harmonic eigenfunc­ tion with eigenvalue A in a graph G (or for an induced subgraph S with nonempty boundary) . Locally, at each vertex, the eigenfunction stretches the incident edges in a balanced way. That is, for each vertex x, f satisfies

L (f(x) - f ( y )) = Aj(x ) d, . II

y �x

Globally, we would like to have some notion that adjacent vertices are close to one another. In spectral geometry, Harnack inequalities are exactly the tools for cap­ turing the essence of eigenfunctions. There are many different versions of Harnack inequalities (involving constants depending on the dimension of the manifold , for example.) We consider the following inequality for graphs:

Harnack inequality:

For every vertex x in a graph f with eigenvalue A satisfies

G and some absolute constant

c,

any eigenfunction

(9. 1 )

Nevertheless, the above inequality i n general does not hold for all graphs. An easy counterexample is the graph formed by joining two complete graphs of the same size by a single edge.

The main goal of this chapter is to establish the Harnack inequality (9. 1 ) for certain homogeneous graphs and their subgraphs . In the next section, we will dis­ cuss several notions of convexity for graphs . After we establish a Harnack inequality for certain homogeneous graphs, we will examine Harnack inequalities for Neumann eigenvalues and Dirichlet eigenvalues for various convex subgraphs. We will also use the Harnack inequality to derive lower bounds for eigenvalues in terms of the diameter and degree of the graph. This section is mainly based on [58] and [56] . 1 35

9 . HARNACK INEQUALITIES

136

9.2. Convex subgraphs of homogeneous graphs Let r denote a homogeneous graph with an associated group 1l and isotropy group I. Let K denote the edge generating set; we require g E K if and only if g - 1 E K. We say that a homogeneous graph is invariant if for every element a E K, we have a K a- 1 = K. In other words, K is invariant as a set under conjugation by elements of K. For example, a homogeneous graph associated with an abelian group is invariant and we call the graph an abelian homogeneous graph. Suppose a homogeneous graph f is associated with a nonabelian group 1£. If the edge generating set K is a subgroup of 1£, then r is still invariant. Before we give definitions for convex subgraphs, some explanations are in order. Although the following various notions of convexity are quite natural and interest­ ing by themselves, they are dictated by the proofs given later for establishing the Harnack inequalities for Neumann eigenvalues and for Dirichlet eigenvalues. An­ other motivation in defining convex subgraphs is in its relevance with random walk problems. Many combinatorial enumeration and sampling problems involve families of combinatorial objects which can be regarded as vertices of convex subgraphs of appropriate host graphs. Harnack inequalities and the resulting eigenvalue bounds can be used to derive polynomial time upper bounds for the rate of convergence for the sampling problem, which often then give efficient approximation algorithms for the associated enumeration problems.

Strongly convex subgraphs An induced subgraph S of a homogeneous graph r with vertex boundary 8S is said to be strongly convex if the edge generating set K satisfies the following condition: ( A) : For all a , b E K , x E 8S, if ax E S, bx E S, then we have b- 1 a E K . We note that (A ) i s implied by the following condition: (B ) : For all pairs of vertices contained in S.

u

and

v

in S , all shortest paths joining

u

and

v

are

Clearly, the intersection of two strongly convex subgraphs is strongly convex. EXAMPLE 9 . 1 . Suppose we consider a homogeneous graph ft with vertex set { (a 1 , a 2 , · · · , at ) : a i E Z , L ai = n} . A vertex (a1 , · · · , ai , · · · , a3 , · · · , at ) is i

adjacent to (a 1 , · · · , ai + 1 , · · · , a3 - 1 , · · · , at ) for 1 � i , j � t. In particular, for a fixed i, we define Hi = { (at , a2 , · · · , at ) : ai � 0} to be a halfplane. It is easy to see that Hi is strongly convex and nHi = { (ai > a2 , · · · , at ) E V(ft ) : ai � 0 } is strongly convex.

1 37

9.2. CONVEX SUBGRAPHS OF HOMOGENEOUS GRAPHS

Convex subgraphs An induced subgraph S of a homogeneous graph said to be convex if for any subset X C 8S, we have

r

with vertex boundary oS is

(9.2)

where N * ( X ) = {y : y E X or y "' x E X } . In other words, any subset X of the boundary 8S of S h as at least as many neighbors outside of S as the cardinality of X . We will call (9.2) the boundary expansion property.

LEMMA 9.2. If two induced subgraphs F1 , F2 subgraph F1 n F2 is convex.

are

both convex, then the induced

PROOF. Suppose X c t5(FI n Fz ) . We can partition X into two parts XI = X n 8F1 and X2 = X \ 8F1 . Clearly X2 is contained in F1 . Since F1 and F2 are convex, we have u

oFI ) J � J X d

U

t5Fz ) J � J X2 J ·

J N * (XI ) \ (F1 J N * (Xz ) \ (F2

Since N* (X2 ) \ F2

F1 U 6F1 , we have

c

(N* (X2) \ ( F2

Hence

J N * (X) \ (F1 n F2

u

U

tS (F1 n F2 ) J

and the proof is complete.

6F2 ) ) n (N* (XI ) \ (F1 > >

U

oFI ) ) = 0.

J N * (XI ) \(FI u t5FI ) J + J N* (X2 ) \ (F2 J X d + J X2 J = J X J .

u

oF2 ) l

0

...

...

EXAMPLE 9.3. We consider the space S of all m x n matrices with non-negative integral entries having column sums c1 , Tm . , C n , and row sums r 1 , First, we construct a homogeneous graph r with vertex set consisting of all m x n matrices with integral (possibly negative) entries. Two vertices u and v are adjacent if they differ at four entries in some submatrix determined by two columns i, j and rows k, m satisfying

j +1 It is easy to see that r is a homogeneous graph with an edge generating set consisting ( 1 - 1 ) halfplane consists of matrices the ( j )th of all 2 2 submatrices 1 -1 Uik

x

=

Vik

+

1, Ujk = Vjk 1, Uim = Vim

1 , Ujm

. A

entry of which is non-negative for some i and j . It is intersection of halfplanes.

= V

m

i,

easy

to see that S is just the

We remark that the definitions given here regarding convex subgraphs are far from being exhaustive. One simple and general notion of convexity is to embed the vertices of the host graph into a manifold. A convex subgraph is just an in duced subgraph on the intersection of the vertex set of the host graph with some subm anifold having convex boundary. The techniques in this section are not enough to derive Harnack inequalities for such a general definition of convex subgraph . We will return to this topic after we discuss the heat kernels of graphs in the next chapter.

9. HARNACK INEQUALITIES

1 38

9.3. A Harnack inequality for homogeneous graphs We will first prove the following Harnack inequality for homogeneous graphs which are invariant. K

THEOREM 9.4. In an invariant homogeneous gmph f with edge genemting set consisting of k genemtors, suppose a function f V ( f ) -+ IR satisfies 1 Lf(x) = k L )f(x) - f(ax)J = >.. f (x) . :

Then the following inequality holds for all x E V ( f} and a > 2 : >..a2 1 k L [f(x) - f(ax)] 2 + a >.. j 2 (x) � a _ sup f 2 (y). 2 y a E /C

� /(.

PROOF.

We define

1 p(x) = k L [f(x) - J(gxW g E /C

and we consider

Lp (x)

=

1 k2 L L { [f(x) - f(ax)] 2 - [f(bx ) - f(abx)] 2} b E /C a E /C 1 - k 2 L L [f(x) - f(ax) - f(bx) + f(abx W b E /C a E /C

+ k22 L L [j(x) - f(ax) - f(bx) + f(abx)][J(x) - f(ax)]. b E /C a E /C

Let X denote the last double summation above. Then

X =

2

k2

L L [j(x) - f(ax) - f(bx) + f(abx)] [f(x) - f(ax) J bE /C a E /C

2 L { L[j(x) - f(ax) - f(bx) + f(bax)] } [f(x) - f(ax)]

k2

aE/C bE/C

+ k22 L [L (f(abx) - J(bax))][j(x) - f(ax)] aE/C b E /C 2 >.. 2 k L [f(x) - f(axW + k L [L (f(abx) - f(bax))][f(x) - f(ax)] . 2 a E /C

Since r is

an

a E /C b E /C

invariant homogeneous graph, we have

L ( f ( abx ) - f(bax)) 0 =

b E /C

and therefore

Lp ( x )

2 >..

�X= k

L [f(x) - f(ax)] 2 .

a E K.

9 . 3 . A HARNACK INEQUALITY FOR HOMOGENEOUS G RAPHS

1 39

Now we consider 1

k

aEK

k

L f (x) [f (x) - f (ax) J - k L [f(x) - f (ax W

2

L [f2 (x) - J2(ax ) J 1

aEK

a E IC

1

2>./2(x) - k L [f (x ) - f (axW .

=

aEK

Combining the above arguments, we have, for any positive 1

L( k

a:

L [f(x) - f (ax)f + a >.j2 (x)) � 2a >.2 P(x)

a E IC

- (a � 2)>. L [f (x) - /(ax)]2 . a E IC

Now we consider a vertex v which maximizes, over all 1

k L [f(x)

x E S, the expression

- f( axW + aA/2(x).

a E IC

We have 0

<

1

L( k

L [f ( v ) - / ( av )] 2 + a .X f2 (v ))

a E IC

< 2a.X2 /2 (v) - >. ( ak 2 ) L [f (v) - f(av)]2 . a E IC

This implies

for

a > 2. Therefore for every x E V ( f ) , we have 1 1 k L [f (x) - f ( ax W + a >.P ( x) < k L [f(v) - f ( av)]2 + aA f2 (x) aEIC

a E IC

2.Xa < a - 2 1 (v) + oAf (x) a2 .X max / 2 ( y ) < -a-2 2

2

Y

for

a > 2.

The proof of Theorem

9.4 is complete.

0

a 4 in Theorem 9.4 we have THEOREM 9.5. In an invariant homogeneous graph f with edge generating set k IC consisting of generators, suppose fu nction f : V(r) -+ IR satisfies By taking

=

a

Lf (x ) = k L [f (x ) - f( ax )] = A f(x ). 1

a E !\:

9. HARNACK INEQUALITIES

140

Then for all x E V (f) , 1

k

2 )f( x ) - f(ax)f � 8A

aE�

sup /2(y). Y

9.4. Harnack inequalities for Dirichlet eigenvalues In this section, we consider Laplacians acting on functions with Dirichlet bound­ ary conditions. Let S denote a convex subgraph of a graph G. We first prove a useful lemma which which is a consequence of convexity.

LEMMA 9.6. For a convex subgraph S of a graph G, an eigenfunction f (with Dirichlet eigenvalue A) defined on S U 8S, can be extended to all vertices of G which are adjacent to some vertex x in S U 88 so that it satisfies

L (f( x ) - f(y)) = Af(x ) dz .

y y�z

P ROOF . First we note that any eigenfunction f must have value 0 for any To extend f to all vertices adjacent to some vertex in S U 88 , we consider a system of I8S I equations :

x E 88.

L (f( x ) - f(y)) = Af(x) dz y y�z for each x E 88. The variables are f (z ) for every z rf. S U 88 and y E S U 88. The boundary expansion condition {9.2 ) implies that any k equations involve at least k variables for k � 1681 and thus assures that this system of equations has z

�

solutions.

0

THEOREM 9. 7 . Suppose S is a convex subgraph in an abelian homogeneous graph r with edge generating set IC consisting of k generators. Let f S -t IR denote an eigenfunction with associated Dirichlet eigenvalue A . Then the following inequality holds for x E S and a E IC : :

for any

o

> 2.

P ROOF. Using Lemma 9.6, we can extend vertex in S U 68. For g E IC, we define

f

to all vertices adjacent to some

c/J9 ( x ) = [f( x ) - f(gx )f + koAj2 ( x )

for x E S U 68. First we note that the maximum value of cp9 (x) , ranging over all g E IC, x E S U 8S, is achieved by some edge { } where z E S. z , az

9.4. HARNACK INEQUALITIES FOR DIRICHLET EIGENVALUES

Let

L4> ( x)

4>{x )

denote

4>a ( x ) .

141

We consider, for x E S,

1

k �)4>(x) - 4> ( bx)]

=

bE.IC

1

<

k L )(f(x) - f(ax ) )2 - (f(bx) - f(abx))2] + o: .\ � )J2(x) - j2(bx)]

=

Y + Z.

bE.IC

bE.IC

We examine Y

= k1 �) (f( x ) IIE.IC

=

- f(ax))2 - ( J( bx ) - f(abx))2]

2 k L (J(x) - f(ax) - f( bx ) + f(abx))(f(x) - f(ax) IIE.IC

1

- k L (f( x) IIE.IC

�

- f(ax ) - f(bx) + f(abx))2

2 k{ L [f(x ) - f (bx) ] - �)/(ax) - f(a bx)]} (J(x) - f(ax)) bE.IC

bE IC

�

2.\[f (x) - f(ax W since Lf( x) = .\f( x ) for x E S U 58. Also, we have z o: >. L:l! 2(x) - f2(bx)] =

<

b E.IC

o: .\ { L 2 [/(x) bE .IC

<

- f( bx) JJ( x ) - L [f(x) - f(bx)]2} IIE IC

o:.\{ 2 k.\f2 (x) - L[f(x) - /( bx )]2} . bEIC

In particular, for v achieving the maximum value of 4>, we have

0 � L4>(z) � 2ko:.\2 / 2 (z) - .\( o: - 2 ) L [f(z) - /(az)]2 a E IC and

2k .\o: [f( z) - f(az) ] 2 � 2 /2 (z ) for > 2. Therefore for all x E S, g E K, gx E S, we have [f (x) - f(gxW + ko:.\ f2(x) < [f( z ) - f (azW + ko:.\f2(x) k.\ < 2 a / 2 (z) + ko: .\ f2(x) o: - 2 k.\o: 2 max < 0: - 2 y E S /2 (y) for any o: > 2. The proof of Theorem 9.7 is complete. a -

o:

By taking

o: 4 in Theorem 9.7 we have =

0

14 2

9. HARNACK INEQUALITIES

THEOREM 9 . 8 . Suppose S is a convex subgraph in an abelian homogeneous graph r with edge generating set K consisting of k generators. Let I : s --* IR denote an eigenfunction with associated Dirichlet eigenvalue >.. Then for all x E S, a E K, we have [f (x) - l ( ax W � 8 k >. sup l2 ( y ) .

yES

9.5. Harnack inequalities for Neumann eigenvalues In this section, we focus on Laplacians acting on functions which satisfy the Neumann boundary condition. For strongly convex subgraphs, a similar but slightly different Harnack inequality for Neumann eigenvalues can be derived. The proof is based on the assumption that the homogeneous graph is abelian and uses the Neumann boundary condition.

THEOREM 9 . 9 . Suppose S is a finite strongly convex subgraph in an abelian homogeneous graph r with edge generating set JC consisting of k generators. Let I : S --* IR denote an eigenfunction with associated Neumann eigenvalue >. . Then the following inequality holds for x E S, a E JC and ax E S:

[l( x ) - l (ax )f

+ kcx>.l2 ( x) � k>.cx22

--

Q -

sup l2 ( y ) yE S

for any ex > 2 . P RO O F .

We consider, for some g E JC, l/J9 ( x ) = [l ( x ) - f ( gx )f + kcx>.l2 ( x ) .

Let { z , az } b e an edge in S at which the maximum value of ¢J9 ( x ) ranging over all g E K, x, gx E S, is achieved. Claim: ¢J9 ( x ) � l/Ja ( z ) for all x, gx E aS U S. Proof. We consider the following three possibilities: Case 1 : Suppose gx E S and x E aS. The Neumann condition implies that

1 l ( x ) = - L l( bx ) W bEW where

w

= ax n s and w = I WI. Therefore we have

l/J9 ( x )

[l( x ) - l( gxW + k cx >.j2 ( x ) 1 1 [ - L l(bx ) - l( gx W + k cx >. [ w w

=

=

bE W

Thus

¢9 ( x )

�

1 "'

-

w

L..... [! ( bx ) �w

- f ( gx )] 2

L f ( bx W .

bEW

+ kwcx>. L f ( bx)2 • -

�w

143

9 . 5 . HARNACK INEQUALITIES FOR NEUMANN EIGENVALUES

From the definition of strongly convex subgraph, bx is adjacent to gx for all bx E Therefore, we have

Case

2:

Suppose

gx E �S and x E S. ¢9 ( x ) = [f(x) - f(gxW + ko:)..P (x ) 1 [f( x ) - L f(bgx W + ko:>.f2(x) =

w

1

<

1

W

1

W

where W' = �g x n S and w because of strong convexity.

3:

Suppose that

b E W'

L [f(x) - f(bgxW + ko: >.f2 (x )

L max 9 ¢19 (x)

bEW'

tPa (z)

<

'

'

bEW'

1

<

Case

W.

=

IW' I ·

Here we use the fact that

bgx is adjacent to x

x, gx E �S. We define Wt {b : bx E S, bgx E S} , {b : bx E S, bgx ¢ S } , W2 {b : bx ¢ S, bgx E S } . W3 =

We set

Wi

¢9 ( x )

IWil and w = w 1 + + Then = [f( x ) - f(gx W + ko:>. f2 (x ) 1 1 = (- f(x) + - "' L..J f(bx) - -f(gx) - -

=

wa .

w2

w2

wa

W

<

W

b E W1 u W,

W

W

"'

L..J b E W, u W3

f( bgx )]2

+ ko:>.P(x) 1 ; { L [f( bx) - f(bgx) ] 2 + L [f(bx) - f(gx )]2 bE Wt

b E W2

+ L [f( x ) - f(bgx) ] 2 }

+ ko:>. { L f 2 (bx) + L f2 (bx) + L f2 (x)}. W bE W2

b E W1

b E W3

Suppose we can show that bx gx for b E w and X bgx for 2 applying Cases 1 and 2, we have ¢19 (x) $ tPa (z) , as desired. �

�

bEW. a

Then by

To see that bx "' g x for b E W , we consider bgx E �S and its neighbors bx , gx 2 are in S. By the definition of strong convexity, we have bx gx. Also, x bgx for b E W3, since bx E �S and its neighbors x, bgx are in S. �

�

The rest of the proof is quite similar to that of Theorem 9.9, so will be omitted. 0

9.

144

By taking

o:

=

HARNACK INEQUALITIES

4 in Theorem 9.9 we have

THEOREM 9 . 1 0 . Suppose S is a strongly convex subgraph in an abelian ho­ mogeneous graph r with edge generating set K consisting of k generators. Let f : S -t IR denote an eigenfunction with associated Neumann eigenvalue A. Then for all x E S, a E K, we have [f(x) - f(axW S 8kA sup f 2 (y) .

yE S

9.6. Eigenvalues

and diameters

The Harnack inequality in previous sections can be used to derive the following eigenvalue inequality: THEOREM 9 . 1 1 . The Neumann eigenvalue A s of a convex subgraph S of an abelian homogeneous graph r satisfies 1 As � 8kD 2 where k is the degree of r and D is the diameter of S. P RO O F . Let f denote can choose f such that

an

eigenfunction defined on

sup l f (x) l

xES

=

88 achieving A s

= A. We

1 = sup f( x ) . xES

Let u denote a vertex with f(u) = maxxES f (x ) = 1 and let v denote a vertex with f (v) < 0. Such a v exists since L f(x) = 0. We now consider a shortest path P xES

in S joining u and v , say, P D, we have t 5 D .

=

(u

We consider S=

=

v0 , v 1 ,

· • •

, Vt

) Since the diameter of S is

= v .

t- 1

L )f(vi ) - f(v;H ) f i=O

S S 8kAD.

By Theorem 9. 1 0, we have On the other hand, we have

t-1

� )f (v;) - f(vi+dF

S > >

Therefore we obtain

and the proof of Theorem

9. 1 1

i=O

1 (f(u) - f(v) ) 2 D 1

n·

1 A > -­ - 8kD 2 is complete.

0

CHAPTER 10

Heat kernels

1 0 . 1 . The heat kernel

of

a graph and its induced subgraphs

Many of the most powerful techniques in spectral geometry involve heat kernels. We will consider heat kernels for a graph or an induced subgraph having Laplacian acting on functions with boundary conditions. The name "heat kernel" is due to the fact that it is a fundamental solution of the following heat equation: 8u - = -Cu. 8t In a way, the Laplacian is associated with the rate of dissipation of heat. To be precise, suppose for a graph G on n vertices, or for an induced subgraph S on n vertices, we consider the Laplacian C acting on functions with Neumann or Dirichlet boundary conditions (as defined in Chapter 8.) We can write C as follows: C=

n- 1

L >..; I;

i=O where I; is the projection onto the ith eigenfunction ¢>; of the induced subgraph S. For any t � 0, the heat kernel H1 of G is defined to be the n x n matrix

H1

= :L e -"'1 I; e-tc

In particular, where

2

i 2 I - tC + 2£ -

I denotes the identity matrix.

. • •

Ho = I

All information about a graph is, of course, contained in its heat kernel. One of the main sources of power in using the heat kernel is the presence of the additional variable t. For example, the it provides flexibility for using the maximum principle. Sometimes, it is easier to establish inequalities or equalities involving the heat kernel either for all t or for a large range of t. We can then often deduce the spectrum by selecti ng t appropriately. We remark that the definition for the heat kernel for graphs is exactly the parallel of the heat kernel for Riemannian manifolds (see the book of Yau and Schoen [ 251)) . The first four sections in this chapter are adapted from [60] . Section 10.5 is based on [79] . 1 45

1 0. HEAT KERNELS

1 46

10.2. Basic facts on heat kernels Let S denote an induced subgraph of a graph G. Let H denote the heat kernel of S subject to Neumann or Dirichlet boundary conditions. LEMMA 1 0. 1 . The heat kernel Ht for a graph G with eigenfunctions ¢>; satisfies

Ht (x,y) :�:::::e-.x• t ¢>; (x)f/>;(y). =

i

PROOF. The proof follows from the fact that

Ht = L e - , t I; .x

i

I; (x , y) = f/>; (x)¢>; (y).

and

0

LEMMA 1 0 . 2 . For 0 $ s � t, we have

L H,(x, y)Ht -a(y , z) = Ht (x , z) . II

The proof of Lemma

10.2 follows from the definition

For a function J : S u oS

-t

(by matrix multiplication) .

lR, we consider

F(x,t)

=

L Ht (x, y) f(y)

11 E SU6S

(Htf)(x) . Here we consider some useful facts about

F and Ht :

L EMMA 1 0 . 3 . (i ) :

F(x, O) = f(x)

( ii) :

Fo

r

x E S U oS,

L Ht (x,y) .jd; = .../d;

11 E SU6S

(iii) :

F

satisfies the heat equation:

8F

= - .C F at (iv) : Under the Neumann boundary condition, we have, for the any vertex x in oS,

.CF( x, t )

=

F t) L ( (x , v'i[; z - 11 II

-

F(y, t ) ) = 0 vcr;

1 0 .2.

BASIC FACTS ON HEAT KERNELS

1 47

For the Dirichlet boundary condition, we have, for any vertex X in oS, F(x, t )

(v) :

j£) - F (�) } 2

=

0

F

= L F (x , t) .C F (x , t) where s· denotes the dll zE S set of all edges with at least one endvertex in S.

L

{ :t , I/ } E S •

PROOF.

(

z

V

(i) is obvious, and (ii} follows b y considering the all 1 's function 1 : II

To see (iii) , we have aF at

= = =

- Hd

a at - .Ce - C t f - .CF.

The proof of (iv} follows from the fact that all eigenfunctions f have a corresponding F which satisfies (iv ) . To prove ( v) , we c onsider

L F (x , t ).CF (x , t )

=

zES

L F(x, t) T- 1 1 2 L T- 1 12 F(x , t)

zES

by using the Neumann or Dirichlet conditio n as described in ( iv) .

LEMMA 1 0 . 4 . For all

x, y

0

E S U oS, we have

PRO OF. The matrix A = I - .C has all entries non-negative. Therefore etA has all non-negative entries. Since all e ntries of H1 are non-negative. The lemma is proved.

0

1 48

10. HEAT KERNELS

10.3. An eigenvalue inequality In this section, we will establish an inequality for lower bounding the eigenvalue >. 1 of a graph or a subgraph with boundary. The main method here is by repeatedly using the heat kernel and using various properties of the heat kernel. Suppose we have a function f : S U 58

g( x, t )

( 10. 1) where

F ( x, t)

=

=

-t R.

We consider

)

(y) F (x, t ) 2 L Ht ( x, y) y'd;il; fIT dz y dy vIT yE S

L Ht ( x , y) f (y) .

(

Therefore, we have

y

(10.2)

g(x , t) =

L Ht ( x, y)Vd:z:/ dyj2(y ) - F2 (x, t )

yE S

B y summing over

x, we have

L g( x , t ) = L L Ht( x, y) Vdz/d11 f2(y) - L F2 (x, t) .

zES

zES yES

z

Using Lemma 10.3 (ii) , (iv) and (v) , we obtain

L g(x, t)

zES

=

'L. f2 (y ) - 'L. F2 ( x, t)

yE S

=

1t d L F2 (x, s)ds

-2

z

O

1 0

ds z E S t d L F ( x , s ) ds F (x, s ) ds zE S

2 10 t L F (x, s ).CF(x, s) ds

( 10.3)

=

We claim that for

Fac t 1 :

L { z ,y } E S "

t ;::: 0, we have

L { z ,y } E S*

( F(x, t) - F(y, t) ) 2 VCf;

To see this, we consider

( 1 0 .4)

2 1O t

zES

04

< -

(x, t) _ F (y , t ) ) 2 ds ( FVCf; . 04 L

{ q} ES*

( f(x) - f(y..;a;) ) 2 VCf;

149

1 0 . 3 . AN EIGENVALUE INEQUALITY

For the Laplacian with Neumann boundary condition, the above expression is equal to

L

2

x E SUoS

!!._ F(x, t ) F(x, t ) ( y'd; dt y'd;

t) t) 2 L !!._ F(x, ( F(x, v'(I; dt v'(I;

=

xES

2L

zES

_

t) _ F(y, ,fif; )

F( y, t)

,fif;

)

!!._ F(x, t) C F(x ' t ) . d t v'(I;

For the Laplacian with Dirichlet boundary condition, ( 10.4) is equal to 2

F(x, t )

L

z E SUoS =

2

L

rES

=

v'd;

F(x, t )

.;a:;

L Y

{ r , y} E S "

L

Y { z , y } E S"

2 L F(x, t) C zES

(

!!._ F(x , t) dt .;a:;

F(x, t ) ( !!._ dt v'd;

_ !!._ F( y , t ) ) dt Vld u.y

_ !!._dt F(y1(4, t) ) y uy

d F(x, t ) . dt

I n either case, ( 10.4} i s equal to 2L xES

=

d F(x , t ) C F ( x, t ) dt

d d - 2 L dt F ( x , t ) · dt F( x, t ) zES

=

<

- 2 "L: (

0.

rES

! F(x , t))2

Therefore

and Fact 1 is proved. Substituting the inequality of Fact 1 into ( 10.3}, we get

L g (x , t)

rES

( 10.5)

10.

150

HEAT KERNELS

In the other direction, we consider the lower bound:

L g ( x , t)

z ES

>

(10.6) Combining

>

(10.5) and (10.6) , we have

Therefore we have proved the following: T H E O RE M 1 0 . 5 .

(i) The Neumann eigenvalue satisfies (10.7)

\

"1

A1

for an induced subgraph

1 "" .

..t f -> yES 2t zL... ES m

Ht (X, y ) ..,fd;

�

with heat kernel Ht

S

with heat kernel H:

.

{ii} The Dirichlet eigenvalue A� for an induced subgraph satisfies (10.8)

S

HHx, y ) ..,fd; A� -> ..!_ inf L 2t z ES yES � .

10.4. Heat kernel lower bounds Theorem 10.5 can be used to establish eigenvalue lower bounds if we can effec­ tively find good lower bounds for the heat kernels. Here, we discuss two examples. The first example is a direct and simple way of using the inequality (10.7 ) . The second example is to bound the kernel for a "convex subgraph" by relating it to an associated continuous heat kernel for an appropriate Riemannian manifold. We consider the special case of a graph G with no boundary. In addition , suppose the graph G has a "covering" vertex x0 with the property that x 0 is adjacent to every other vertex y in G. The degree of x0 is n - 1 where G has n vertices. We will apply ( 10. 7 ) with t --+ 0. Since

H1

=

I - t£ + O(t 2 ) ,

10.4. HEAT KERNEL LOWER BOUNDS

lSI

it follows from ( 10. 7) that >

_.!_ "" inf 2 t L yES zES

Ht (x, y) ..(d;

uyy yfd

( (;J

)

1 t inf + O(t2 ) 2 t 1/oFZo dy 1 = 2t where 62 denotes the second l argest degree i n G. Thus, we have 1 (10.9) A I :::: . 262 For example, for G = P3 , the path with 3 vert i ces , it is true that A0 = 0, A 1 = 1, and A2 = 2, while our estimate in ( 10.9) gives A 1 2:: 1/2. A pp lyi n g this to G = Kn , the complete graph on n vertices, yield s n ,\ > 1 - 2 (n - 1 ) while the true value i s A 1 = n/ (n - 1) (again off by a factor of 2 ) . I n fact, as pointed out by L. Lovasz, A1 2:: 1 /62 follows by directly using the Rayleigh quotient . =

For the remainder of this section, we will restrict ourselves to special subgraphs of homogeneous graphs that are emb ed ded in Riemannian manifolds. Such a re­ striction will allow us to derive ei gen alue bounds for graphs us i ng the known results on eigenvalues of Riemannian manifolds. Also, the res t r ic ted classes of graphs still inc lude many families of graphs which arise in various app l i cat ions in enu me ration and sampling. We remark that in Chapter 9, we derive eigenvalue lower bounds for subgraphs of homogeneous graphs with stronger convexity conditions by using a Harnack inequality. Both the d efi nit ions and the meth ods are different here. Roughly speaking, our goal here is to use the ( onti nuous ) heat kernels of the manifolds with convex boundary for deriving as the lower bound function for the (discrete) heat kernel.

v

c

Suppose r is a h omogeneou s graph with associated group H. Here we assume that the edge generating set K c 1i is symmetric, i e , g E K if and only if g - 1 E K .

..

Suppose vertices of r can be embedded into a manifold M wi t h a measure J-1 such that for any

J-t (x , gx )

g

=

J-t (

y , gy )

E K and x, y E V ( f). I n addition, if J-t (x , g x )

= J-t (y , g 'y )

.

for any g, g' E K and X , y E V(f) then r is called a lattice gra p h An induced subgraph on a subset S of ver ti ces of a lattice gr aph r is sai d to be a convex u bg aph if there is a submanifold M with convex bo u n d a y such that S consists of all vertices of r c on tai ned in M, i.e.,

s r

S = M n V (f) .

r

Furthermore, we require that for any vertex x, t he Voronoi region Rx J-t(y , z ) for al l z E r n M } is containe d in M.

=

{y

: J-t( y ,

x)

<

10.

152

HEAT KERNELS

EXAMPLE 1 0 . 6 . As in Example 9.3., we consider a homogeneous graph r with vertex set consisting of all m x n matrices with integral (possibly negative) entries , Cn , and row sums r1 , . . rm . Two vertices x and y are having column sums c1 , adjacent if they differ at four entries in some submatrix determined by two columns i, j and two rows k, m satisfying . • .

.

X;A: = Yik + 1 , Xj k = Yi k - 1 , X;m = Yim - 1 , Xjm = Yim + 1 It is easy to see that

( -� -� )

r

all 2 X 2 submatrices

is a homogeneous graph with edge generating set consisting of . Obviously,

r

can be viewed as being embedded

in the mn-dimensional Euclidean space M = !Rmn . In fact, r is embedded in a (mn - m - n + I )-dimensional subspace of M. We consider the set S of all m x n matrices with non-negative integral entries with given row and column sums. Suppose we choose the submanifold M , determined by

Xij

>

1

-2.

It is easy to verify that S is a convex subgraph of the lattice graph r.

Here we state the main theorem for bounding the eigenvalue A 1 of a convex subgraph. THEOREM 1 0 . 7 . Let S denote a convex subgraph of a lattice graph and suppose S is embedded into a d-dimensional manifold M with a convex boundary and a distance function J..I. . Let JC denote the set of edge generators and suppose t: = min{ J..I. ( x, gx) : g E JC} . In the first order approximation of the discrete Laplacian L by the continuous Laplace operator, ( 10 . 10)

2d IJCI

L: (

g E JC •

(_

J..I. ( x, gx)

)2

� 8g 2

82 a; ; _E , 8x •· 8x}· i,j where JC• consists of exactly one of a and a- 1 for all a E JC . Suppose that C1 I � (a;,; ) � C2 I where I is the identity matrix. Then the Neumann eigenvalue A1 of S satisfies the following inequality:

where

U lSI

r = vol M '

KERNEL LOWER BOUNDS

10 . 4 . HEAT

U de n otes the volume of the Voronoi regi o n, Co �

for some absolute

153

and eo is an absolute constant s atisfying

Co min {C1 , C2 1 }

constant C0 .

The proof of Theorem 10.7 is rather long and complicated . We will ske tch the major ideas here ; the details can be found in (60] . T h ere are three major parts.

Part 1 : Lower bounds for the continuous heat kernel. Let h(t, x, y) denote the heat kernel of M and let u(t, x) = h(t, x, y) sat i sfy the

heat equation

a

( � - at )u ( t , x ) = 0 with the Neumann boundary condit i on ov

a

for any

boundary point x.

u(t, x) = 0

Here the Lap l ace operator � is taken to be (2 a2

�=I KI L fi2 g · g E K.

Also, we assume that

JJ.(x, gx) = l for all

x E

V (f) and

g E K.

Li and Yau established the following lower bound for [184] :

H

i n their seminal paper

Theorem: Let M de n ot e a d-dimensional compact manifold with boundary OM . Suppose the Ricci curvature of M is no nn egative , and if 8M =/: 0 , we assume further that aM is convex. Then th e fu n dam e nt al solution of the heat equation with the Ne um ann boundary condition satisfies , y) h(t, x, y) � C - 1 (t) (vol ( Bx ( vr;t ) ) - 1 exp -( 4JJ. ( xl)t _

for some constant C ( l ) depending on f > 0 and d (Here, vol( ) denotes volume and Bx (r) denotes ball of radius r ce n te red at x.) ·

such that C(l) the intersection

-t oo

0. as with the

of M

f -t

However, the above version of the usual estimates for the heat kernel cannot b e direc tly used for our purposes here since the constant C is exponentially small dep ending on d. A more carefu l an alysis of the heat kernel is needed . To lower­ bound the discrete heat kernel, we will use the following lower bound estimates for the ( co ntinuous ) heat kernel (a proof can be found in [60] ) .

10 .

154

For any o:

>

0, and

u

HEAT KERNELS

� cda,

( 1 + o:) -d - ( 1 + o:)�J-2 (x, y) exp 4B., (y'(it) at Suppose we choose o: = � , and c2 t0 = dD 2 (M) , where c2 = 2f2 fd and D (M) denotes the diameter of M . (We may assume D(M) � 1 . ) By using (10. 1 1 ) we have (10. 1 1 )

h ( t , x, y ) �

Part 2: Approximate the discrete heat kernel by the continuous heat kernel. We define the following function k (t , x, y) which will serve as a lower bound for the heat kernel of the graph. k(t, x, y )

=

Ct

JM

h ( c2 t, x - z , y) cp (z) dz

where cp is a bell-shaped function, for example, a modified Gaussian function exp( -c' l z/E I 2 ) with compact support, say, { lzl < E/4}, and which satisfies (10.12)

Ct

j cp(z)dz =

c3 (c4f)d

where c3 and c4 are chosen so that the above quantity is within a constant factor of the volume of the Voronoi region R., . So, k(t, x, y) can be approximated by h(c2t, x, y) U or ( 10.13) when t is not too small (using the gradient estimates of h which are also proved in [60] ) ) . Here U denotes the maximum over x of the volume of R., . From ( 10.12) , we have k(to - E1, x, y)

=

> >

Ct

JM

Ct ct>

vol M C7 U " vol M

h(c2 (to - E 1 ) , x - z , y)cp(z)dz

JrM cp(z)dz

where C7 � c' min { c1 , C:J 1 } and the Harnack inequalities in the continuous case are used. We note that the c's (with the exception of c2 , which is a scaling factor) denote some appropriate absolute constants . In the next section, we will give an example of computing the c's in the eigenvalue computation of Example 10.7. In order to show that k(t, x, y) is indeed a lower bound for H1 (x, y ) , a proof is needed. There are several sets of sufficient conditions for establishing lower

1 55

1 0 . 4 . HEAT KERNEL LOWER BOUNDS

bounds of Ht (X, y) . The detailed proofs in (60] are somewhat long and will not be reproduced here. Using this, we then have Ht (x, y ) � vol M

crU

-- .

Part

3:

Eigenvalue lower bound.

Now we can combine all the estimates and use Theorem 10.5. Hence, ,X

2 I>mf Ht (x, y) 1

>

t x E S yE S

cr U l S I

>

t vol M cr t: 2 r fP D(M)2

>

where U denotes the volume of the Voronoi region and r denotes the ratio U I SI /vol M. This completes the sketch of the proof for Theorem 10.8. To get a simpler lower bound for .X , we note that the diameter D(S) of the convex subgraph S and the diameter of the manifold are related by ( 10. 14)

D(M) :$ t: D(S).

Therefore, we have the following:

COROLLARY 1 0 . 8 . Let S denote a convex subgraph of a simple lattice graph and suppose S is embedded into a d-dimensional manifold M wi th a convex boundary. Then the Neumann eigenvalue .X 1 of S satisfies the following inequality: eo r .X t � fP D2 (S)

wh ere

U l SI vol M ' D (S) denotes the (graph) diameter of S, K, denotes the set of edge generators, and Co is an absolute constant depending only on the si mp l e lattice graph. r

=

REMARK 1 0 . 9 . For a polytope in JRd , we can rescale and ch oose the lattice points to be dense enough to approximate the volume of the polytope . For example, if we have :$ D1 (M)/d

wh ere D 1 denotes the diameter of M measured by the L1 nor m and C is some ab sol ute constant, then the number of lattice points provides a good approx im at i on for the volume of the polytope. This implies that r 2: c for some constant c. The above inequality ( 1 0 . 1 5) can be replaced by a slightly simpler inequality: ( 10. 1 5)

C

t:

C ' d :::; D(S) for some constant C' . These facts are useful for approximation volume of a convex body, which we discuss in t he next section .

algorithms for the

10.

156

HEAT

K ERNELS

REMARK 10. 10. There are marty graphs G that cart be embedded in a lattice graph in such a way that the diameter of G satisfies

D (G) ,..,

Vd D(M) . t

For such graphs, Theorem 10.7 implies a somewhat stronger result:

A� where r is as defined in Theorem 10. 7.

eo r

tPD(G)2

1 0.5. Matrices with given row

and

column sums

One of the classical problems in enumerative combinatorics is to count ( or es­ timate ) the number of matrices with nonnegative entries having given row and column sums ( as described in Example 10.7) . This problem arises in a variety of applications, such as graph enumeration , goodness of fit tests in statistics, enu­ meration of permutations by descents, describing tensor product decompositions, counting double cosets, etc. In particular, in statistics it is often referred to as the "contingency table" problem artd has a long history. It seems to be a particu­ larly difficult problem to obtain good estimates for the number of such tables with given row and column sums. In order to attack this problem , a standard technique depends on rapidly generating rartdom tables with nearly equal probability. In this section, we will use the theorems in the preceding sections to estimate the eigenvalue A 1 for the convex subgraph with vertices corresponding to matrices with given row and column sums. By using these bounds we can derive convergence bounds for Neumann random walks on this subgraph [79] .

( r 1 , . . . , rm ) , c = (c1 , . . . , en ) with r; , Cj � 0 and Given integer vectors r E Cj , we consider the space of all m x n arrays T with non-negative integer

E r; i

=

=

j

entries satisfying with the property that

L T (i, j) Let us denote by T

=

r; ,

l �j�n . =

T(r, c) the set of all such arrays.

We will show that for the Neumann walk on the space of tables T(r, c) where min min � . min

we have

( 10. 16)

{

•

n

1

ci }

m

>

c ( m - 1 ) 312 ( n - 1 )31 2

1 0 .5.

157

MATRICES WITH GIVEN ROW A N D COLUMN SUMS

We first need to place our contingency table problem into the framework of the preceding section. The manifold M will consist of all real mn-tuples x = (x u , X12 , Xmn) satisfying • .

.

,

L X;; = r; , L Xij = c1 . i

j

Since E r; = E c; then i i

dim M

=N

(m - 1 ) (n - 1) .

The graph r has as vertices all the integer points in M , i.e., all X with all Xij E z . The edge generating set JC consists of all the basic moves described above. Hence, IJCI = (';') (�) . The set S will just be T = T(r, c) , the set of all T E r with all entries nonnegative. Thus, :=

S = n;,; {T E f : x;; � 0} .

Similarly, the manifold M

M

M is defined by

= n;,; {x E M : x;;

C

>

- 1/2} .

It is clear that M is an N -dimensional convex polytope and S = T is the set of all lattice points in M, and consequently convex in the sense needed for ( 10.10) . It is easy to see that T is connected by the basic moves generated by JC, and that each edge of r has length 2. Our next problem is to deal with the term ����o�r in (10. 10) . In particular, we would like to show this is close to 1 , provided that r; and c; are not too small. To do this we need the following two results. Claim 1 .

radius of L

(

)

N JR. is a lattice generated by vectors v1 , 1 /2 N is at most R : = � IJ v; 112

Suppose L

c

i�

..

.

, vN .

Then the covering

PROOF . The assertion clearly holds for N = 1 . Assume it holds for all dimen­ sions less than N. It is enough to prove that any point x = (x1 , , x N ) in the fundamental domain generated by the v; is at most a distance of R from some v; . Let x0 be the projection of x on either the hyperplane generated by v 1 , , vN _ 1 , or a translate of the hyperplane by VN , whichever is closer (these are two bound­ ing hyperplanes of the fundamental domain) . Thus, d(x,xo ) � � llvN II · By the induction hypothesis,

..

.

.. .

for some j < N . Hence,

as

claimed.

0

158

1 0 . HEAT KERNELS

Claim 2. IT M is convex and contains an open ball B (eRN) of radius eRN, with eN � 2, then

4-I /c

( 10. 17)

where

V1 ' . . . VN

< l S I vol U < 1 - vol M -

generate r, and R =

1 /2 N ) ( � v�1 ll vi ll 2

PROOF. Consider a contracted copy ( 1 - 6) M of M with the origin centered about the center of the ball B(eRN) C M. Let L be any bounding hyperplane of M, and let (1 6)L be the corresponding contracted copy of L.

-

Let x E ( 1 - 6) M and suppose there exists a lattice point y E V (f) \ S such that x is contained in the Voronoi region for y. Hence, R > d(x, y) �

1

� e6RN .

However, this is a contradiction if we take 6 = c}v .

Consequently, for the choice 6 = c}v we must have Voronoi region of some lattice point in S. Therefore,

x

in the closure of the

l SI vol U � vol ( 1 - 6)M

i.e.,

l SI vol U > vol M if eN � 2 . Claim 2 is proved.

( 1 - _1_ ) N > 4- 1 / c -

eN

0

In order to apply the result in Claim 2 , we must find a large ball in M . Let so denote the smallest line sum average, i.e. , so

=

.

(

.

Ti

•

n

.

ej) .

mm mm - , mm -

We begin constructing an element T0 loss of generality that

E

1

m

M recursively

as

follows. Suppose without

TJ

so = - . m

Then , in T0 , set all elements of the first row equal to s0 , and subtract so from each value Cj to form cj = ei - so , 1 :::; j :::; n. Now, to complete T0 , we are reduced to forming an ( m 1) by n table T� with row sums r 2 , r m and column sums c� , . . . , c� . The key point here is that all the line sum averages for T� are still at least as large as s0 . Hence, continuing this process we can eventually construct a table T0 (with rational entries ) having least entry equal to s0 . Consequently there is a ball B ( s0 ) of radius s0 centered at T0 E M which is contained in M (since to leave M, some entry must become negative) . Therefore, if we assume so > cN312 then by Theorem 10.7 and ( 1 0 . 1 7) ,

-

( 1 0. 18)

• . •

4 eo4- I /c

A I � N2 (diam M ) 2

for some absolute constant Co > 0 where eN � 2 (since for tables, all the generators have length 2, so that R :::; N 1 12 ) .

10.5. M ATRICES WITH GIVEN ROW AND COLUMN SUMS

1 59

At the end of this section, we illustrate how a specific value can be derived here for eo ( as well as in several other cases of interest, as well) . In particular, for contingency tables, we can take eo = 1/800. Since

d;am M < 2 mm

{ (�>f', (�A)"'}

then (10. 18) can be written as follows:

For the natural Neumann walk P on the space of tables 7(r, c) where

{

Cj } >

; min min r , min

we have

•

n

m

1

c ( m - 1 ) 312 (n - 1 ) 3 12

( 1 0. 1 9)

To convert the estimate in ( 10.16) to an estimate for the rate of convergence of P to its stationary ( uniform ) distribution 1r , we consider Ll(t)

<

(1

_ Ad

. vol S

mm degr x z

( 10. 2 0) <

e-�� t

vol S deg r

=

e

->.t iS I

.

and At is the least nontrivial Neumann eigenvalue of Laplacian of

S.

Thus, if (10. 2 1 )

then Ll (t) < f . Note that

I TI

� m;n

Thus, by ( 1 0 . 16) and ( 1 0 .2 1 ) , if ( 10. 2 2)

t > 6400e ' l ' m ' n ' m;n then Ll (t) < f , provided

{

{ r.r r.r cj } '� ·

{ �,1, �cj } H { � �In c, }) + mm n

· n

c· n

r , min ..1. min min ....!. •

1

}>

Jn ,, , m

c ( m - 1 ) 3/2 (n - 1) 3/2 .

160

10. HEAT KERNELS

It remains to bound the constant Co occurring in ( 1 0 . 10) . Briefly, as in Theorem

1 0 . 7 , we have

2d

IKI

LJC ·

gE

(

" (x , gx)

l

{)2

)

2

!:..._ 8g 2

L a •; ox Bx · i,j

=

1

1

min l'(x, gx) , I' denotes (Euclidean) length, and gE IC consists of exactly one element from each pair {g, g- 1 } , g E K. where

d

: = dim M ,

f. : =

K*

C

K

Suppose G1 and G2 are constants so that

where

I is the identity operator on M , and X $ Y means that the operator Y - X is In particular, we can take for G1 and G2 the least and greatest

non-negative definite.

eigenvalues, respectively, of ( a1; ) restricted to M . Now , from the arguments in it follows that when

M

to be

[60] ,

is Euclidean then the constant Co in (10. 10) can be taken eo =

�

1 0

min (Gl > G:2 1 ) .

Thus , to determine Co in various applications, our job becomes that of bounding the eigenvalues of the corresponding matrix ( a;; ) . First, we consider

x;;

- x;';

- X ij ' + X i' i'

m x n

contingency tables. With each edge generator g = 82 � m terms of t he x ' s . •

we cons1" der

Expanding, w e have

We can abbreviate this in matrix form as

X ij X i' j X ij ' X i' j ' 1 -1 -1 1 X ij Xi' j - 1 1 1 - 1 Xij ' - 1 1 1 - 1 1 X i' j ' 1 - 1 - 1 We need to consider the operator

10.6. RANDOM WALKS AND TH E HEAT

KERNEL

161

The corresponding matrix Q has the following coefficient values for its various entries: Entry Coefficient

(Xij , Xij) (Xij , Xi' j ) (Xij , Xij' ) ( Xij , X i' j' )

(m - 1 ) (n - 1 ) - (m - 1 ) - (m - 1) 1

Thus, Q has two distinct eigenvalues: one is mn with multiplicity (m - 1)(n - 1 ) , and the other i s 0 with multiplicity m + n - 1 . Now, dim M = ( m - 1 ) (n - 1} and the operator corresponding to Q when restricted to M has all eigenvalues equal = to mn. So the matrix ( a ij ) has all eigenvalues equal to 8, and

2mn1r:;.){��n-l)

consequently we can take C1 = C2 = 8, and eo = 1 /800. This completes the computation of eo in the lower bound of .X 1 for the contingency table problem. In [79] , using similar methods, eigenvalue lower bounds are derived for a num­ ber of problems including restricted contingency tables, symmetric tables, compo­ sitions of an integers, and so-called knapsack solutions.

10.6. Random walks and the heat kernel In the previous sections, we discussed methods for bounding eigenvalues using the heat kernel. As we will see here, heat kernels also play a crucial role in bounding the rate of convergence for random walks in a direct way. We will consider the relative pointwise distance (as defined in Section 1 .5) to the stationary distribution for a random walk with an associated weighted graph G.

THEOREM 1 0 . 1 1 . Suppose a random walk is associated with a weighted graph G with heat kernel H1 • The relative pointwise distance of the random walk of t steps

to the stationary distribution is bounded above by

vol G Ht (x, y ) r;r-;:r 6 (t) � mz ax. ,y y d x d11 P ROOF. Let P denote the transition probability matrix of the random walk. We note that the convergence of the random walk p • after s steps is related to the heat kernel H. as follows:

i;tO (10. 2 3)

where we recall that eigenfunction.

l;

is the projection onto the eigenspace generated by the i-th

1 0.

1 62

HEAT KERNELS

After s steps, the relative pointwise distance of p• to the stationary distribution 1r( x ) is given by j P • (y , x ) - 7r ( x) l �(s ) = max _ z, y 1r( x) Let '1/Jx be defined by if y = x , '1/Jx ( Y ) = otherwise. We have lt/JI/ (P t ) '1/Jx - 7r(x ) l � ( t ) = max .:._:'---'-----'--':--:------' � x, y 1r ( x ) l t/J r - 1 / 2 ( I - _C ) t T 1/ 2 '1/Jx - 7r( x ) l = max �11�------'�--�-':-----'�--�� z ,y 1r ( x ) 1 max l t/J11T- 1 2 (L: Jf)r- 1/ 2 '1/Jx lvolG x

{�

,y

( 10.24)

i#O

<

max lt/J11T- 1 1 2 (Ht - Po)T- 1/ 2 '1/Jx l volG z

<

max Ht ( x , y)

,y

x ,y

Hence, if we choose t such that

vol G

ITT

v dx dy

vol G , e minx dx then, after t steps, we have � ( t ) $ L In the next section, we will use Theorem ':01 Gdz in 10. 1 1 and the logarithmic Sobolev techniques to reduce the factor of minz 1 t = - log A

( 10.24) .

D

Theorem 10. 1 2 implies that an upper bound for the heat kernel can be used to bound the convergence rate for random walks. This and many other powerful consequences of the heat kernel wil l be discussed further in the next two chapters.

CHAPTER 1 1 Sobolev inequalities

1 1 . 1 . The isoperimetric dimension of a graph In spectral geometry, Sobolev inequalities are among the main tools for control­ ling eigenfunctions. In this chapter, we will derive the discrete versions of Sobolev inequalities. The proofs are self-contained and entirely graph-theoretic although many of the concepts and ideas can be traced back to the early work of Nash [251] on Riemannian manifolds. On one hand, graphs and Riemannian manifolds are quite different objects. Indeed, many of the theorems and proofs in differential geometry are very difficult to translate into similar ones for graphs (since there are no high-order derivatives on a graph) . In fact, some of the statements of the theorems in the continuous cases are obviously not true in the discrete setting. On the other hand , there is a great deal of overlap between these two disparate areas both in concepts and methods. Some selected techniques in the continuous case can often be successfully carried out in the discrete domain. In particular, we will see that the methods of Sobolev inequalities for differential manifolds can be effectively utilized in dealing with general graphs. In Chapter 9, Harnack inequalities have been derived only for very special graphs (homogeneous graphs and their convex subgraphs) . Here we will establish Sobolev inequalities for all graphs by using a key invariant, the isoperimetric dimension of a graph (also see [59] ) . The proof techniques in this chapter are quite similar to their parallels for the continuous case (49] , [251] .

We say that a graph G has isoperimetric di m e n si o n 8 with isoperimetric con­ stant c0 if for every subset X of V (G) , the number of edges between X and the complement X of X, denoted by I E ( X, X ) l , satisfies (11.1)

-

&-1

I E(X, X ) I � c.s (vol X ) .-

where vol X � vol X and c6 is a constant depending

only

on

8.

In other words, we can define the so-called Sobolev constant c6 by co

=

I E ( X, inf X (vol X ) .-

�)l

where vol X � vol X . We will prove two discrete versions o f the Sobolev inequalities i n Sections 1 1 .2 and 1 1 .3. We will show that for 8 � 1 and for any function J : V (G) ---+ IR, 1 63

1 1.

1 64

SOBOLEV INEQUALITIES

( 1 1 .2)

and for 8 > 2, ( 1 1 .3)

where 'Y = /!.2 • Here the constants c 1 and c2 depend only on 8. In fact we will > r.::- o - 1 o-1 S hOW c 1 > C0 -- an d C 0 2 y C0 21J · _

_

The Sobolev inequalities can be used to derive the following eigenvalue inequal­ ities ( see Section 1 1 .4): ( 1 1 .4)

'"' -.x.t L..... e • i# O

vol G

:S c3 � t

and ( 1 1 .5)

for suitable constants c3 and c4 which depend only on 8. In a sense, a graph can be viewed as a discretization of a Riemannian manifold in IRn where n is roughly equal to 8. The eigenvalue bound in ( 1 1 .5) is an analogue of Polya's conjecture [21 1] for Dirichlet eigenvalues of regular domains M in IRn : 2 /n 27r Ak

>

- Wn

(-k- ) vol M

where Wn is the volume of the unit disc in IRn . Here, we consider Laplacians of general graphs and obtain eigenvalue estimates in terms of the isoperimetric di­ mension. We remark that a closely related isoperimetric invariant [48] is the Cheeger constant ha of a graph G ( discussed in Chapter 2 ). In fact, the Cheeger constant can be viewed as a special case of the isoperimetric constant c6 with 8 = oo . We note that the first Sobolev inequality ( 1 1 .2) can be expressed as

IIV f lh and the second one ( 1 1 .3) as

2: c 1 1 1 ! 1 L b

IIV f ll 2 2: c 2 ll f ii 'Y

for 'Y = 28/ (8 - 2) . In general we can define the Sobolev constants for p, q > 0, as follows:

Sp ,q

. f sup /# 0 c ER

= m

sp ,q

of a graph,

IIV / II p II! - C I . q

For example, the Cheeger constant ha of a graph G is just s 1 , 1 . The two Sobolev inequalities in ( 1 1 .2) and ( 1 1 .3) concern s 1 , q and s 2 , q for certain q depending on the isoperimetric dimension 8.

1 1 . 2 . AN ISOPERIMETRIC INEQUALITY

165

There is another concept related to isoperimetric invariants, the so-called "mod­ erate growth rate" or "polynomial growth rate" condition (15] ( also see (98]}. For a vertex v in a graph G and an integer r, we define

Nr ( v )

=

{u E V (G)

where d( u, v ) denotes the distance between nomial growth rate of type ( c, d) if

: u

d(u, v ) :=:; r }

and

v.

We say a graph G has poly­

vol Nr (v) ?: c r 0

for all vertices

v

and vol N. :::; vol Gf2.

It can be shown that a graph with isoperimetric dimension 8 and isoperimetric constant c0 has polynomial growth rate of type ( c' , 8) for some c' . However, there are graphs which have polynomial growth rate of type ( c, 8) but which do not have isoperimetric dimension 8. For example, consider the graph H which is formed by taking two copies of a graph having polynomial growth rate of type (c, 8) and joining them by an edge. It is easy to see that H has polynomial growth rate, but H cannot have isoperimetric dimension 8. In fact, this is an example which has polynomial growth rate, but its associated random walk is not rapidly mixing. On the other hand, graphs with bounded isoperimetric dimension have good eigenvalue bounds and are therefore rapidly mixing. We remark that for symmetrical graphs, these two notions -polynomial growth rate and isoperimetric dimension- are basically the same. Using a result of Gromov (148] on the growth rate of a finitely generated group, Varopoulos [244] showed that a locally finite Cayley graph of an infinite group with a nilpotent subgroup of finite index has polynomial growth rate of type ( c, 8) depending only on the structure of the group. An excellent survey on this topic can be found in [1 5] .

1 1 . 2 . An isoperimetric inequality

We will first prove the following. THEOREM 1 1 . 1 . In a connected graph G with isoperimetric dimension 8 > 1 and isoperimetric constant c0 , for an arbitrary function f : V (G) -+ JR, let m denote the largest value such that

L

v f(v) <m

Then

wh ere c 1

dv :=:;

L

u f(u) ?. m

du .

L if(u) - f ( v ) i ?: ci (L i f ( v ) m j 6 dv ) !.? =

o-

l. c0 -0-

v

-

Here we state two useful corollaries. The first one is an immediate consequence of Theorem 1 1 . 1 and the second one follows from the proof of Theorem 1 1 . 1 .

166

1 1 . SOBOLEV INEQUALITIES

CoROLLARY 1 1 .2 . In a connected graph G with isoperimetric dimension o and isoperimetric constant c0 , for an arbitrary function f : V (G) --t IR we have

L 1 / (u) - f (v) l � c1 �n ( L 1 /(v ) - m l 6 dv ) Y v u�v

where

c1

=

1

c0 °6 •

COROLLARY 1 1 . 3 . In a connected graph G with isoperimetric dimension o and isoperimetric constant c0 , for a function f : V (G) --t IR and a vertex w , define fw (v) -

_

where

m

{ mmin{ ax { f(v) , f( w) } f (v) , f (w) }

if f(w) � m if f(w) < m

is as defined in Theorem 1 1 . 1 . Then

L l fw (u) - fw (v) l + aw l f( w) - l � ci ( L fw (v) 1�l ) Y m

v

where

C1

=

Co 0-,

o-1

l { { u , v } E E (G) : J (u) > f( w) � f (v) } l l { { u , v } E E ( G ) : f (u) � f( w ) > f(v) } l Sw

-

_

{ {u{ v :: f(v) � f(w) } J( u ) :::; f(w) }

if f( w) � if f (w) <

if f(w) � if f( w ) <

m m.

m

m.

Roughly speaking, the proof of Theorem 11.1, although a bit lengthy, is just repeated applications of summation by parts (which, of course, is just the discrete version of integration by parts) . Namely, for any x; and y; , for i = 0, · · · , n + 1 , the following simple equality holds: n n x; )y; = L x ; (Yi - 1 - Y ; ) + Xn+ I Yn - XoYo ( 1 1 .6) L (XH1 i=O

1

and the last two terms vanish if xo = Yn = 0 or Yo repeatedly in the following proof of Theorem 11 . 1 :

PROOF.

For a given function f

satisfy

:

V(G)

=

Xn+ 1

=

0 . We will use ( 1 1 .6)

lR, we label the vertices s o

as

to

� f(vn ) · f (v1 ) � f(v2 ) � { { vj , vk } E E(G) : j :::; i < k } and a ; = I A; I . We use the f (v; ) and d; = dv ; · Define s;- = L di st = L di and · ·

We define A ; notation f( i )

--t

=

·

'

S; = m in { S;- , St } Clearly, S; = St for f ( i) � From the definition in ( 1 1 . 1 ) , we have .

a;

We use the convention that suppose f ( i o ) = m.

S0

=

m

j
and

� c0 (vol S; ) ---. .

j�i

S;

=

S;-

for f ( i ) <

m.

&-1

Sn+ l

=

0. Let h ( i )

=

h (v; )

=

f(v; ) -

m,

and

167

11.2. AN ISOPERIMETRIC INEQUALITY Then

L i h(u ) - h(v) l

L a;lh(i - 1 ) - h(i)l 6-1

c.s L: si--. (h(i - 1) - h ( i))

>

iio

W2 can be dealt with in a similar way. 6 -1 .S-1 6 - S; 6 ) W1 c.; L h(i)((S; + d;) i$ io 8-1 . d > c.; L h(i) 8 Sl: i$io 6 1 '"" h(i) ;r::r d; 8 > c.; -- L 8 i$io (h(i) _6_ 6-1 S; ) l /6 0 L lh(i)l o - 1 d; 8 - 1 i c.; -1 /o 8 lh(i) l d� 1 d; o •S• - .S- 1 8---1 lh(i) l -" "-1 d;) o . 2: c6 8 L i$io

We consider W1 first ;

=

)

(�

Combining this with the corresponding bound for W2 , we have w, +

w,

=

>

c,

6

"' 6

�1 �1

and Theorem 1 1 . 1 is proved.

((�

lh(i) l '� ' d;) . , .

(� lh (i) l '�' d;) , .

.

+

(,t: lh(i)l '�' d/6'

) 0

We remark that Corollary 1 1 .3 follows from the above proof and the fact that for f(w ) � m ,

L i fw (u) - fw ( v) i

=

L a; (h(i - 1 ) - h ( i) )

i /(i ) ?_/(w)

168

1 1 . SOBOLEV INEQUALITIES

L

>

i h(i)�h(w)

J h(i) J (a; - a;_ I ) - aw J h(w) J .

1 1 .3. Sobolev inequalities

The second Sobolev inequality ( 1 1 .3) is more powerful than the first one ( 1 1 .2). However, its proof in Theorem 1 1 .4 is also more complicated than that of Theorem 1 1 . 1 . Basically, the proof consists primarily of iterations of "summation by parts" using Theorem 1 1 . 1 .

THEOREM 1 1 4 . For a gmph G with isoperimetric dimension 6 > 2 and isoperi­ metric constant C& , any function f : V (G) -7 JR. satisfies .

where

'Y =

6�2 and c2

PROOF.

and f (io)

=

=

,;ci6;/ .

We follow the notation in Theorem 1 1 . 1 where h(i) = h(v; ) m. Also, we denote h; (x) = hv, (x) .

=

f(v; ) -m

L J h(u) - h(vW =

,L: (h(i - 1 ) - h(i) ) L:

Jh(j) - h( k ) l

,L: (h(i - 1 ) - h(i)) L:

J h; (j) - h; ( k ) l

{j,A: } E A;

>

i $ io

{j, A: } E A ;

,L: (h(i - 1 ) - h (i)) L: J h; (j) - h; ( k ) l i�io {j,A: } EA; P1 + P2 . +

=

We will give a lower bound for the second part P1 where without loss of generality we may assume that all h(i) 's are positive. The second part can be proved similarly. P1

>

L: h(i)

i :5 io

> >

L: h(i)

i:5 io

( (

L:

J h i+1 U ) - h i+1 ( k ) 1 -

ai + l (h (i) - h (i

+ 1n -

L: {j,A: } E A ; - A i + t

L h (i) Gi + t (h(i) - h(i + 1) ) - Pl .

i :5 i o

L:

J h; ( j) - h; ( k ) l

{j,l: } E A ;

{j, A: } E A ; + t

J h; (j ) - h; ( k ) J

)

)

11.3. SOBOLEV INEQUALITIES Now we use summation by parts again and we obtain

2P1

>

L h(i)ai+ 1 (h(i) - h(i + 1) )

>

1$io L h(i)(( ai+1 - ai)h(i) - a;(h(i - 1 ) - h(i))

i $ io

> >

L h(i)(ai+1 - a;)h(i) - L h(i)(ai+ 1 (h(i) - h(i - 1) ) i$ io i$io L h(i) (ai+1 - a;)h(i) - 2Pl i$io

We define

T;

=

L ai (h(j - 1) - h(j)) + a; lh(i) l. j$i

From Cor. 1 1 .3, we have

T;

L lfv, (u ) - fv, (v) l + a; (f(i) - m ) _6_ 6-1 6-1 6 � C6 h(i ) 6-i dj ) 6 - (L i
•

It follows from the definition of T; that

T; - T;-1

= (a; - a;-1 )h(i - 1 ) .

169

1 70

11.

Hence, P1

>

SOBOLEV INEQUALITIES

� i:5io L h(i)(ai+l - ai)h(i)

i < io

In

a similar way we have

Therefore,

(�, -h(J) 2) (h(i)

,...,,

since 'Y � 2 .

1

j

� v'C66

� 1 (� l h(i)l "' d; ) � '

0

1 1 .4. Eigenvalue bounds

In this section, we assume a > 2 since we will use the second Sobolev inequality to establish eigenvalue bounds. Let H1 denote the heat kernel of G. From Section 10.2, the heat kernel satisfies Ht (X, y )

=

L H, (x, z)Ht-a (z, y) .

11. 4 .

EIGENVALUE BOUNDS

1 71

In particular, we have Ht ( x , x )

L (H! (x , y ) ) 2 .

=

II

We consider, for a fixed

x,

a H (x, x) at t

=

a 2 "' � Ht ( x, y ) - H !. (x , y ) at 2 2

II

II

since the function H t2 ( x , y ) , with y as the variable and x fixed, satisfies the heat equation in Lemma 10.3 ( iii ) . Now we use Lemma 10.3 ( iv ) and get

� Ht ( x , x ) _- _ � "' at

y �z

(

H � (y, x)

I'd

_

H� ( z , x)

V ""ll

We apply Theorem 1 1 .4 by again considering H t2 (y, x ) x. For 'Y = 6"2_! , we have 2

a ( «5 - 1) 2 H (x, x ) :5 -C6 48 2 at t

( 1 1 . 7)

To proceed, we need an additional fact. LEMMA

(� (

v'd;z as

Ht (y, x)

Ja;

1 1.5.

� + �· L:: /; g; :::; (L:: Jr)lfp(L: unl/q

P ROOF. We apply Holder's inequality for 1

where we take p =

'Y

- 1, q =

� and

=

i

Ht (x, y)

J11 = I Ja; - m l >-• , Ht (x , y) - ml U11 = I 2 We then have

.,fd;

-L-

� ,_,

·

)

2

a function of y with fixed

-

) ) m d11 "�

2h

1 1 . SOBOLEV INEQUALITIES

1 72

It remains to bound

LI

( x , y ) - mldy from above. "Ja; dy

H.

y

By using Lemma 10.3 ( ii ) , we have

L H ; ( x , y ) Jdy - m' vol G

( 1 1 .8)

!I

..jd; - m'vol G 0

= =

by choosing

m

'

to be

m

Define Ni

LI y

H

=

�y ) dy

{y :

H ! (!l z) 17 y d.

I

.;cr;

=

vol a ·

� m ' } and N;

=

{y :

H! (y ,z) 17 y d.

<

m' } . Now

- m' l dy =

<

2 L m'dy yE N;

= 2

v� · L d11 yE N;

<

2 -./d;.

Therefore,

LI y

H

�y) dy

- mldy

< <

H

�y) -

m' l dy + L l m' - ml dy y y y 2 -./d; + L m 'dy y

LI

1 1 .4 . EIGENVALUE BOUNDS

1 73

provided that m' � m/ 2 > 0. This, however, follows from the definition of m as in Theorem 1 1 . 1 and the fact that H � 0 and can be deduced as follows: m ' = ..[(I; vol G

�

� >

where M:

==

{y

:

� vo�

vo G

)

L H � ( x , y ) .Jd; + L H! ( x , y ) .Jd; yEM;

yEMt

L H � ( x , y ) .Jd;

yE Mt m vol M: vol G m 2

H!- ( 11 z ) rT y dv

G

(

�

m

} and M;

==

{y :

H! (y , z ) rT y dv

<

m

} . Recall from the

definition of m , we have vol M: � vol G /2. This completes the proof of Lemma 1 1 .5. 0 We note that

because of { 1 1 .8) . We now return to inequality ( 1 1 .7) . Using Lemma

11.5, we obtain

4

1 1 . SOBOLEV INEQUALITIES

17

where c5 = (c.s <";6V2 ) 2 h - l ) h . We note that 1 - 2( � ) = - � - We consider dz 2 a )( ( , ) at Ht x x - vol G 1 dz 1 2 ll=!l a - � , 8t ( Ht ( x x ) - vol G ) dz 2 =! a ) - 2 ( � ) t Ht ( x , x ) = - 8 ( Ht ( x , x ) vol G a 2 5 ( )+ 2 > � ( Ht ( x , x) - " ) - � 2 ( � ) ( 3 ../d:) - 2 ( �) v G 2 5 ../d: - 2 ( � ) > � (3 )

�

using the preceding inequality for dz

d

z 2 U-=-� ) es (3 v/J dz ) - � t + ( Ho (x , x ) - vol G ) - 1 > es ( 3.,fd;) - 2 < � >t + (1 - �)-i vol G > es (3.,fd;) - 2 < �> t = 2c5fo. This i s equivalent to 2

) ( Ht ( x , x ) vol G - 1

where C6

ft Ht ( x , x) . Therefore, we obtain

2 6 .. Hence, where c3 = 9cd/

Since

>

" L. Ht (x, x) - 1 z

CJVOl G t 'I

-6- .

L Ht ( x , x) = L ( L e - -'• t � (x )) = L e - -'• t ,

z we have proved the following: THEOREM 1 1 .6 .

(1 1.. 9 ) where c3 is

�

a

z

i

-X, � c3v�l G L e- t t 'i i;J!O

constant depending on

/j ..

From Theorem 1 1 .6, we derive bounds for eigenvalues

..

THEOREM 1 1 . 7 . The k-th eigenvalue A.1; of C satisfies

) 2 /d A � c4 ( .1; vol G k

where c4 is

a

constant depending on

PROOF. From ( 1 1 .6) we have

/j ..

c3 vol G k e - -'• t < - t!

2

l l.5.

GENERALIZATIONS TO WEIGHTED GRAPHS AND SUBGRAPHS

for all t > 0. Therefore

"' e •t c3 vol G inf -6t t2 .A ,�:e 6 2 = CJVOl G . ( --) 2 {J is minimized when t = 2 t . This implies <

k

since the function

*

1 75

.A .�:

·

>

fJ

k

2

( ) 2e c3 vol G 1 k 12 = c4 ( vol G ) 0

We remark that the constants c3 and c4 can be explicitly computed from the proofs but are somewhat messy expressions. Here we derive these inequalities with no intention of carefully optimizing the constants. 1 1 .5 . Generalizations to weighted graphs and subgraphs

In this section, we consider weighted undirected graphs G with edge weights = Wv,u for vertices u , v of G. We can define the isoperimetric dimension of G in a similar way : We say that G has isoperimetric dimension {J and isoperimetric constant cc� if L L Wu,v � c.s (vol X) 1 - t Wu,v

uEX

vE X

for all X � V(G ) with vol X � vol X where dv

=

L Wu, v , and vol X = L dv . u

vE X

The results in previous sections can be generalized for the Laplacians of weighted undirected graphs with boundary conditions. We will state these facts but omit the proofs which follow along the lines in the previous sections. THEOREM 1 1 . 8 . Let G be a weighted undirected graph with edge weights W u, v for vertices u, v of G, and suppose G has isoperimetric dimension {J and isoperi­ metric constant c6 • Let S denote a subset of vertices of G . Then, any function f S U {JS ---t 1R with either Dirichlet or Neumann boundary conditions satisfies 6 el - l 6 L l f(u ) - f(v ) lwu ,v � c 1 m�n ( L l f ( v) - ml c!-I dv ) :

v ES

{u, v }ES•

where s• consists of all edges with at least one endpoint in S, and c1 c.s 6u1 . THEOREM 1 1 . 9 . Let G be a weighted undirected graph with edge weights for vertices u, v of G, and suppose G has isoperimetric dimension {J > 2 and isoperi­ metric constant c6 . Then any function f V (G) IR with either Dirichlet or Neumann boundary conditions satisfies L l f ( u ) - f(vWwu,v ) 1 12 � c2 m�n ( L I J (v) - m i "Y dv ) 1 h =

Wu , v

:

{ u,v}ES•

---+

vES

1 76

II.

where "f =

2 .! 6_2

and c 2 =

SOBOLEV INEQUALITIES

r,;-; c! - 1

y C.S 'ilJ ·

THEOREM 1 1 . 1 0 . For a weighted undirected gmph G and an induced subgmph S, the Dirichlet or Neumann eigenvalues of S satisfy CJVO} S "" e - ). . t � · �� t io/ 0

where

c3

is a constant depending only on b .

THEOREM 1 1 . 1 1 . Suppose a weighted undirected gmph G has isoperimetric di­ mension b and isoperimetric constant c0 . Let S denote an induced subgmph. The k-th Dirichlet or Neumann eigenvalue of S satisfies Ak � C4 ( v k 2 / 6

ol G )

where c4 is a constant depending only on b .

CHAPTER 12 Advanced t echniques for random walks on graphs

1 2 . 1 . Several approaches for bounding convergence

Suppose in a graph G with edge weights h as transition probability

w., , 11 ,

a random walk (x0 , x1 ,

•

• •

, x. )

v P(u, v) = Prob (xi+1 = v I Xi = u) = Wu,

d,.

where d,. =

� Wu, v · The number of steps required for a random walk to converge v

to the stationary distribution

1r,

rr ( x ) =

where d.,

vol G '

vol G = "' � d., , z

is closely related to the eigenvalues of the Laplacian (see Sections 1 .5 , 10.5) . In particular, suppose we define .\1 if 1 - .\ 1 � An-1 - 1 .\ (12.1 ) 2 - A n - 1 otherwise. Then the relative pointwise distance of P to 1r after s steps satisfies

_{

A(s)

if

vol G c s � \ log . d

( 1 2.2)

.

m1n., ., " We remark that this can be slightly improved by using the lazy walk as described in Theorem 1 . 16. Therefore, the .\ in ( 12.2) can be taken to be (12.3)

.\ =

{

1

2 1

An - 1 + .\ 1

otherwise

Here .\; 's are eigenvalues of the Laplacian of the graph G ( see definitions in Chapter 1) .

d G In this chapter, we will describe several methods for reducing the factor log m':01 •n in ( 1 2 .2) by using the log-Sobolev constant. •

Suppose G = (V, E) is a graph with edge weights w.,,11 for x, y E V. The log­ Sobolev constant o of G is the least constant satisfying the following log-Sobolev 177

•

1 78

1

2

.

ADVANCED TECHNIQUES FOR RANDOM WALKS ON GRAPHS

inequality for any nontrivial function f : V

� lR:

L (f (x } - f (y)) 2 wr, y � L f 2 ( x}dr log a zEV

{r, y} E E

In other words,

(12.4}

a

£

x ) vol G f 2 ( z }dz

zE V

can b e expressed as follows: a

a =

a

=

. f

L ( f (x ) - f ( y Wwz ,11 { z ,y } E E

P (x )vol G

f 2 ( x ) dz log }�o rL L f 2 (z ) dz EV zEV

where f ranges over all nontrivial functions f : V

� JR.

Logarithmic Sobolev inequalities first arose in the analysis of infinite dimen­ sional elliptic differential operators. Many developments and applications can be found in the survey papers [16, 149, 1 50, 231] . Recently, Diaconis and Saloff-Coste [101] used a discrete version of the logarithmic Sobolev inequality for improving convergence bounds for Markov chains. In Section 12.2, we will improve ( 12.2) to: Ll (t) � e2-c if vo!G c 1 t ?. � log log . d + \ . ( 12.5) mln;r ;r A 2 Also we will show Llrv (t) � el-c if vo!G 1 c ( 12.6) t ?. . d + ,A . 4 log log mln z z This is a slight improvement of a(101] in which Diaconis and Saloff-Coste proved that Llrv (t) � e l -c if 1 c t ?. a log log n + :X 2 for regular graphs. So , lower bounds for log-Sobolev constants can be used to improve convergence bounds for random walks on graphs and certain induced sub­ graphs. However, the problem of lower bounding log-Sobolev constants seems to be harder than finding eigenvalue bounds ( even for special graphs ) . One relatively easy method is to use comparison theorems that will be discussed in Section 12.2. ..__.

A direct and powerful approach for estimating the log-Sobolev constant is by using the logarithmic Harnack inequalities. In Section 12.4, we will establish a logarithmic Harnack inequality which has a similar flavor to the Harnack inequality (discussed in Chapter 9) . This provides an effective method for controlling the behavior of the function achieving the log-Sobolev constant. However, the proofs here hold only for homogentous graphs and their special subgraphs.

1 2 . 1 . SEVERAL APPROACHES FOR BOUNDING CONVERGENCE

179

Another approach for bounding the convergence rate for random walks in a graph is by using Sobolev inequalities and the isoperimetric dimension of a graph ( defined in Section 1 1. 1 ) . This will be discussed in Section 12.5. As a consequence , for graphs with bounded isoperimetric dimension, their random walks have partic­ ul arly nice bounds for convergence. The definition for the log-Sobolev constant for a graph G can be easily gener­ alized to induced subgraphs with boundary conditions. Let S denote a subset of vertices in a graph G and let s• denote the set of edges with at least one endpoint in S. Let 6S denote the vertex boundary of S. The log-Sobolev constant akD) for the induced subgraph S with Dirichlet boundary condition can be defined as follows: (12 . 7)

where f ranges over all nontrivial functions f : S U 6S

y

E 68.

-t

IR satisfying f (y)

Also, the log-Sobolev constant as for the induced subgraph boundary condition can be defined as follows:

=

0 for

S with Neumann

(12.8)

where f ranges over all non-constant functions f : S U 8S -t JR. Many methods for bounding log-Sobolev constants for graphs can be extended to the log-Sobolev constant for certain subgraphs as well. A relatively easy upper bound for the log-Sobolev constant is half of the first eigenvalue .A 1 of the Laplacian C of G. LEMMA 1 2 1 For a gmph G (with no boundary or with Dirichlet or Neumann boundary conditions), we have .

.

(12.9) PROOF. Let f denote a harmonic eigenfunction associated with eigenvalue .A1 . We consider f ' = 1 + t. f for some small f. > 0. Since Lz f (x) dz = 0, we have

/';<'\.../ ... . ; �· :�

1 2 . ADVANCED TECHNIQUES FOR RANDOM WALKS ON GRAPHS

1 80

From the definition of a (see ( 12.4) ) , we have

L a

<

(J ' ( x) - J ' ( y) ? w.,,11

{ z , y} E E

I II

zEV

We consider I=

L

(J' (x) - J' (y) ? w ., ,11 =

t

{z,y} E E

Also, by repeatedly using

2

L

(J (x) - f ( y)? w, ,11•

{z, y } E E

= 0 and log(1 + x) =

L f (x) d

,

x - x

2

/2 + O(x3 ) , we

z

have

z

By taking

t

close to 0, we obtain )q . a< - 2

D

There are many examples which show that equality in ( 12 .9) does not hold [98] . Suppose ( 12.9) holds or even that the log-Sobolev constant is within a constant factor of .X 1 . Then the random walk on G converges to its stationary distribution 1 in order O(.X! log log(vol G /max degree)) . For a regular graph on n vertices, the 1 number of steps required is of order .X! log log n. For example, for the n-cube Qn, it can be shown that its log-Sobolev constant is equal to >. 1 /2 = 1/n. Therefore the lazy random walk on the cube Qn ( with 2 n vertices) converges in time order n log n. We remark that just as the eigenvalues of a weighted cartesian product of graphs can be determined (see Section 2.6) , i.e., .X a r;w =

1

2 ( >.a + >.H) ,

the log-Sobolev constants also also have the same property: 1

aG® H = 2 (aa + aH ) ·

This can b e used t o determine the log-Sobolev constant for Q n and will b e left an exercise. 1 2 . 2 . Logarithmic Sobolev inequalities

as

Let G denote a weighted graph on n vertices. For a function f : V (G) --t IR, we may view f as a column vector, or a 1 x n matrix as a row vector. The stationary distribution 1r(x) = d., fvol G will be regarded as a column or row vector. Let 1f denote the diagonal matrix with value 1r(x) = dz /vol G as its (x, x)-entry.

1 2. 2 . LOGARITHMIC SOBOLEV INEQUALITIES For a function f

,. 11 / 11,, to be

:

181

V(G) --+ R, we define the ("rr ; p)-norm of J, denoted by

,. 11 / 11,

=

(

L JP(x)tr(x) z E V(G)

)

1 /p

In particular,

The main proof for ( 12.5) consists of two parts. In the first part (Theorem 12.2) , we will see that the inequality (12. 10) relating the p-norm to the 2-norm for certain p, implies the improved convergence bound for random walks. The second part (Theorem 1 2.4) states that the inequality (12. 10) can be derived from the log­ Sobolev inequality. fies

THEOREM 1 2 . 2 . Suppose that in a weighted graph G, its heat kernel

H.

satis­

(12. 10)

for all f : V (G) --+ 1R, and p = e /36 for some positive value {3 . Then the random walk on G satisfies

if

c 2 volG t � 7.i log log d + \

min., "' .

�-'

"'

where A is as defined in {12.1) (or {12. 3) for the modified walk defined in {1 . 1 6)). P ROOF. We define

q

by 1 1 - + - = 1.

p

q

For a vertex x of G, let t/J., denote the characteristic function satisfying t/J., (y) if x = y, and 0 otherwise. For a function f : V --+ R, we consider (12. 1 1 ) =

<

=

l t/1., 'Tr - 1 / 2 H;rr i / 2 /I I t/J., 'Tr-1+ 1 / q ( 'Tr l / p -1/2 H.'Trl/2 f) I

(

(�

)

(,P,

1r-' (y) )

'

•(Y)) (� ' I•

,. llt/J., 1r - 1 ll q ,. ll/ 7r l /2H. 1r - l /2 11 ,

by using Holder's inequality.

(f1r'i' H, 1r - 'i' (y))'n(y)

)

' ' '

=

1

1 82

1 2 . ADVANCED TECHNIQUES FOR RANDOM WALKS ON G RAPHS

We consider

. u �.rr -' 11,

�

�

(�(�.

rr - ' ( y ) ) ' •(•l

('rr ( x )1 - q ) 1 fq 1r ( x) -1jp �olG ) 1/P. ( mmz dz

)'

''

Using the hypothesis that p = ef3• and choosing s as volG 1 , s={3 log log

.

min z dz

we have

( �olG ?IP minz dz

=

From (12.10) and (12.11) , we have, for

l t/Jz 71"- 1/ 2 H. 71"1/2 / I

ee

'os l os

mTD'!�a - fJ •

=

e.

any / , � e " 11 / 71" 1 / 2 H. 71" -1 / 2 l i P � e , 11/112·

In particular, for the heat kernel and the proj ection /0 into the zeroth eigenfunction, we have l t/Jz 71"-11 2 (H•+ r - Io ) 11" 11 2 !I < l t/Jz 71" -11 2 H. (Hr - Io ) 11"1/ 2 / I < e , 11 71"-11 2 (Hr - Io )71"11 2 / 1 ! 2 <

< <

This is equivalent to

e ll (Hr - Io ) 71" 1/2 /1 ! 2 e ii (Hr - Io ) ll2 11 71"1 1 2 /112 e1 - .Xr " 1 1 / 112·

for all g. This implies (12.12) Therefore, the random walk on G converges to the stationary distribution under relative pairwise distance as follows (see (10.23)) :

71"- 1 1 2 (H2 a+ 2 r � (2s + 2r ) < max z l t/Jz

- Io )71"- 1 / 2t/J11 1 < max llt/Jz 71" - 11 2 ( H.+r - Io ) l l 2 · l l t/Jy 71"- 1/ 2 (H.+r - Io ) ll2 z ,y < e2 - 2 .X r by using ( 12.12) and the Cauchy-Schwarz inequality. Now, we take r 2\ ,y

2s + 2r, and the proof is complete.

=

,

t

=

0

We can also obtain a similar st atement for the convergence bound under the total variation distance

.

1 2 .2 . THEOREM

fies for all f

:

1 83

LOGARITHMIC SOBOLEV INEQUALITIES

1 2 .3. Suppose that in a weighted graph G, its heat kernel H, satis­

V(G)

,. ll f -+ IR,

walk on G satisfies

7r l/ 2 H,

and p

if

=

7r -1/ 2 11 P �

e fl • for some positive value (3. Then the random

volG 1 t� (3 log log . d + m1n., .,

,.. ll f ll 2

c

\ "

where >. is as defined in {12. 1) {or {12.3) for the modified walk as in {1 . 1 6}). P ROOF .

We follow the notation in Theorem 12.2. 1 llrv = 2 m;x L 1 1/Jx p• + r ( y ) - 1r ( y ) l II

<

� m;x L 1 1/Jx II

< <

1r- 1 1 2 ( Hs + r -

Io)7r 1 / 2 ( y ) l

1 r - max "' .t e 1 - � 1r ( y ) 2 z L... II 1 1 - �r 2e

by using (12.12).

0

(3

Now we proceed to show that the log-Sobolev constant can be used to determine in the above theorems. This proof is very similar to the continuous case ( [149] ; also s ee [101 ]) . THEOREM 1 2 . 4 . In a graph G with log-Sobolev constant a , its heat kernel Ht

satisfies

for any t > 0, P ROOF .

p=

7r 1 1 2 H 7r- 11 2 IIP � ,. ll f ll t ,.. ll f 2 e401 + 1 , and for any f : V(G) -+ JR.

From the definition of a , we have

}: U( x ) - f ( y )) 2 w ( x , y ) ;:::

z-11

a

L f2 ( x) dx log "'

JZ ( x ) Z

2 L....t f (z)1r(z)

x

z

for any nontrivial function f. In particular , we can replace f by f'P/ 2 and we obtain ( 1 2 . 1 3)

}: U PI2 ( x ) - JPI2 ( y ) ) 2 w( x, y ) ;:::

z -11

a

}: fP ( x ) dx log x

f ( x )P

L jP(z)1r (z) z

Now we need the following inequality which is not hard to prove: (12. 14) 4(p - 1 ) ( aP 1 2 - bP/ 2 ) 2 � p2 (a - b) (aP - 1 - bP- 1 ) .

.

1 2.

184

for all

a,

b

ADVANCED TECHNIQUES FOR RANDOM WALKS ON GRAPHS

� 0 and p � 1 . From (12.13) and ( 12.14) , we have a L JP(x) 7r (x) log z

f (x)P L JP(z)7r(z) z

(12. 15)

<

4

: 1) L (JP- 1 (x) - JP- 1 (y))(f (x) - f (y))w.,,11 z -y

We now repl ace f by g function of t: Note that p'

=

p' (t)

=

f 7r11 2 Ht 7r-11 2 in the above inequality and define p as a P

=

=

p ( t)

=

1 + e4a t .

4a {p - 1 ) . From (12. 15) , we have

( 12.16)

g(x) I P � L 9P(x)7r (x) log L lgP(z)1r(z) L(gP-I (x) - gP- I (y))(g(x) - g(y))w.,,y � 0. - z11 z

z

Now we define

F(t)

=

..

IIYIIp·

Clearly, F(O) = ,. 11! 11 2 . If we can show that the derivative F' (t) � 0 , then we have ,. llg ll p = F (t) � F(O) = ,. 11111 2 as desired. It remains to show F' (t) � 0. Since F(t)

we have

�

(� (f1T''' H,1T- ''' (•))'•(x)) '

' ' �

FI (t)

(12.17) We note that G' (t)

=

i (G(t)) ' • ,

( - p2p' log G(t) + pGG'(t) ) F(t) . (t)

p L gp-I 7r (x) ( f 7ri /2 d Ht1r-If 2 (x)) + p' L gP(x) 7r (x) log g(x) t z z d

=

I + II

We consider the above sum of I transpose of A) : I

=

= =

p 9 p -1 7r l /2

as

a product of matrices (where

! Ht 7r l /2 r

A* denotes the

gp - I 1r i /2c_ Ht 1r l /2 f * -p gp - 1 7r l /2 c_7r l /2 g • P _ "" (gP- l (x) 9p - I ( y)) (g(x) g( y))wz ' 11 vol G L..J -p

_ _

_

z-y

_

1 2 . 3 . A COMPARISON THEOREM

FOR

THE LOG-SOBOLEV CONSTANT

1 85

by using the heat equation in the weighted version of Lemma 10.3 . Substituting into ( 12. 17) , we obtain p' F' (t) p2 (L gP (x) 1r (x) log gP (x) - log G(t)) z

- 1 L (gP- l (x) - gP - l ( y)) (g (x) - g(y)) vol G z - y

::::;

0

by using ( 12.16 ) . This completes the proof of Theorem 12.4.

0

Therefore we have THEOREM 1 2 . 5 . A(t) ::::; e 2 - c if

In

a weighted graph G with log-Sobolev constant

1 2o:

t � - log log THEOREM 1 2 . 6 . Arv (t) ::::; e 1 -c/2 if

In

1

-

4o:

log log

we have

o:,

we have

c vol G + \. . mm., d., "

a weighted graph G with log-Sobolev constant t�

o: ,

c vol G + \. . mm., d., "

1 2 . 3 . A comparison theorem for the log-Sobolev constant

An improvement of the convergence bounds in ( 12.4) depends on knowing (or estimating) the value of o:, which, if anything, is harder to estimate than A1 (for general graphs ) . We can bypass this difficulty to some extent by the following ( companion ) comparison theorem for o:. Its statement ( and proof) is in fact quite close to that of Lemma 4.14. LEMMA 1 2 . 7 . Suppose G = (V E ) and G ' = (V', E') a re connected graphs, with logarithmic Sobolev constants a = aa and a ' = aa• , respectively. Suppose P : V --t V' is a surjective map satisfying: ,

( i ) If d., and d�, denote the degrees of v E V and x' all x' E V ' we have

(ii ) For each edge e

E

d., � cd�

•

E

V', respectively, then for

.

z E p - 1 (z' )

= x y E E there is a path P(e) between such that: ( a) The number of edges of P ( e ) is at most i;

p(x)

and p(y) in E'

1 2.

1 86

ADVANCED TECHNIQUES FOR RANDOM WALKS ON G RAPHS

(b) For each edge e'

E E' , we have

L

We

e:e' E P(e)

where

We

:5 m We' .

denotes the weight of the edge e. Then a ' � lm a . c

(12. 18}

PROOF. Consider a function g : V'

stant

a

'

in G' . Define f : V --+ a

'

IR

--+ IR which achieves the log-Sobolev con­ such that f (x) = g(p(x)) . Then we have

L g2 (e ')we'

e' E E'

=

S(g) L g2 (e ')we'

e' E E'

( 12. 19)

L: J2 (e)w.

L J2(e )w.

eEE

I x ll x iii . eEE

=

S(J)

S(f) . S(g)

where S(g) = L f 2 (x)d., log

.,

g(x)

L P (z)1r(z) . z

Exactly as in the proof of Lemma 4. 14, we obtain

I � lm ' II � a .

1

It remains to show III � cf. [101] ) . First, define

c

(which we do using a nice idea of Holley and Stroock;

F( � , ( ) := � log � - � log ( - � + (

for all �' ( > 0. Note that F(� , ( ) for some eo > 0,

�0

:tE V

�

.

>

.

L

:r' E V'

>

This implies

III �

= c

c

and for ( > 0, F(�, ( ) is convex in

L:;. ) .

.

,

d,

{.

Thus,

F(g(x')' )

cd�, F(g(x' ?) since F � 0

L

F(g(x' ? d�, ) by convexity

z' E V '

cS(g) .

and (12. 18) is proved.

0

HG

12.4. LOGARITHMIC HARNACK INEQUALITIES

187

and G' are regular with degrees k and k' , respectively, then we have k > aI k'f.m a . -

1 2 .4. Logarithmic Harnack inequalities

In previous sections we discuss log-Sobolev techniques that were adapted from the continuous case. In this section, we will consider the logarithmic Harnack inequality which was first motivated by the discrete problem on random walks. As it turns out, we can obtain the logarithmic Harnack inequalities for both Riemannian manifolds and for finite graphs. For a smooth, compact, connected Riemannian manifold M , we let V denote the gradient with the associated Laplace-Beltrami operator �- Suppose M has no boundary or has a boundary which is convex (as defined in (12.5) ) . We consider Ll acting on functions f : M -+ IR satisfying

j 1! 1 2 = vol M

(12.20)

satisfying the Dirichlet or Neumann boundary conditions. The log-Sobolev constant a of M is the least value satisfying: (12.21)

JM f2 ( ) log f2 (x) � a JM IV f(x) l2 • x

Suppose f is a function achieving the log-Sobolev constant a. Then it can be shown that f satisfies the following equation: �� -a f log f 2 (12.22) Using ( 12.22), the following logarithmic Harnack inequality for the function f de­ fined on M satisfying ( 12.20) : ( 1 2.23) I V fl2 + a f 2 log j 2 � as up(J 2 Iog f 2 ) . =

provided M has non-negative Ricci curvature.

The inequality in (12 .22) is similar to the Harnack inequality except for a logarithmic factor. So, we call (12.22) the logarithmic Harnack inequality. It can be used to derive the fol lowing lower bound for log-Sobolev constants for a d­ dimensional compact Riemannian manifold M with non-negative curvature ( also see (96]) : ( 12.24)

where D(M) denotes the diameter of M and .X1 is the first eigenvalue of the Lapla­ cian. Suppose S denote an induced subgraph of G . We consider all nontrivial func­ tions f : S u �S -+ IR satisfying :z:

1 2 . ADVANCED TECHNIQUES FOR RANDOM WALKS ON G RAPHS

188

where dz denotes the degree of x . The log-Sobolev constant o with the Neumann boundary condition satisfies:

where f ranges over all nontrivial functions satisfying the Neumann (or Dirichlet) boundary condition. (The case with Dirichlet boundary condition can be worked out in a similar way.) The function f achieving the log-Sobolev constant o in ( 1 2 .4) satisfies: ( 1 2 .25)

} )f ( x) - f (y)) = odzf (x ) log j 2 ( x )

II �� �z

where y ,...., x means y is adjacent to x . For the discrete case, we can only establish the logarithmic Harnack inequality for invariant homogeneous graphs (which are defined later in Section 3) .

For a graph G with isoperimetric dimension t5 (see definition in Section 1 1 . 1 ) and with the assumption that G is a k-regular invariant homogeneous graph or a strongly convex subgraph, we can use (12.26) to show that ( 12. 2 7)

where D denotes the diameter of G. Since a random walk on a graph G on n vertices is close to stationarity after order log log n/o steps, the above lower bounds for the log-Sobolev constant o immediately imply a convergence bound of order (log log n) kD 2 if the isoperimetric dimension is bounded. The proofs for log-Harnack inequalities are somewhat complicated. We will give a proof to ( 1 2 . 25) and describe both the discrete and continuous results. THEOREM 1 2 . 8 . For a graph G, suppose f : V -+ IR satisfies {12. 20) and achieves the log-Sobolev constant, and L P ( x ) dz = vol G . Then f satisfies, fo r any vertex x ,

Lf (x) =

L ( f(x ) - f(y)) o f( x) dz log P ( x) . =

II

�� �z

12.4. LOGARITH MIC HARNACK INEQUALITIES

1 89

PROOF. We use Lagrange's method, taking the derivative with respect to f (x} of the log-Rayleigh quotient (which is the right side of ( 12.4} ) . Then we have 2 (/ (x} dz log f 2 ( x) + 2 f ( x } dz } L ) f (x ) -

2Lf(x)

( 12.28}

for some constant fied to:

c1 •

f(y))2

After substituting for a, the above expression can be simpli­

L f ( x ) - a ( f (x) log f 2 ( x} + 2 f ( x ) ) + c z f (x ) = 0.

( 12.29}

After multiplying ( 12.29} by f(x) and summing over all x in V, we have

L: U (x } - f(y )f - a 2: f2 (x } (log f 2 ( x) + 2) + c2 L f 2 (x ) = 0.

This implies

c2

=

2a. Th e refore we obtain from ( 12.29}

L f (x)

=

a

f ( x ) dx log f 2 ( x ) .

0

We state here the following logarithmic Harnack inequality for homogeneous graphs which are invariant (as defined earlier i n Chapter 9) .

THEOREM 1 2 . 9 . In an invariant homogeneous graph f with edge generating set K consisting of k generators, suppose a function f : V(r) lR achieves the log-Sobolev constant and satisfies {12.20}. Then the following inequality holds for all x E V (r) : 2: !J(x) - f(ax)f � 6ka max{U2 log U2 ( 1 + � log2 U2 ) , 1 } --+

a E IC

where

U

=

sup j f ( y ) j � 1 . y

THEOREM 1 2 . 1 0 . In an abelian homogeneous graph f with edge generating set consisting of k generators, consider strongly convex subgraph S of r. Suppose a function f : S U 8 S --+ lR satisfies the Dirichlet or Neumann boundary condition and achieves the log-Sobolev constant. Also, assume 2: f 2 (x)dx = vol S . Then xES the following inequality holds for all x E S : L: U(x) - f(ax W � 6ka max { U 2 log U 2 ( 1 + � log2 U2 ) , 1 } .

K

a

a E IC

The proofs of Theorems 12.9 and 12.10 are s i mi l ar to, but somewhat more complicated than, those in Sections 9.3, 9.4, and 9.5. The complete proofs can be found in [62] .

There are several useful properties that a function achie vi n g t h e lo g-Sob olev constant possesses. Here we state some without giving the proofs (see [62] ) .

12.

1 90

ADVANCED TECHNIQUES FOR RANDOM WALKS ON GRAPHS

THEOREM 1 2. 1 1 . In a connected invariant homogeneous graph r with edge generating set K; consisting of k generators, suppose a function f : V (r) --+ lR achieves the log-Sobolev constant and L P (x )d:r = vol G . Then for all x E V (r) , we have

U = sup l f (y) l $ e k . II

where e is the base of the natural logarithm. THEOREM 1 2 . 1 2 . In a connected invariant homogeneous invariant graph G = (V, E) , suppose a function f : V --+ lR satisfies the logarithmic Harnack inequality and L f 2 (x)d:r vol G. Then the log-Sobolev constant o of G satisfies :r 1 1 . ) 0 � mm ( 32kD2 ' 24kD2 log U2 ' =

where U

=

supz 1/(z) l , k denotes the degree, and D denotes the diameter of G .

T H EO R EM 1 2 . 1 3 . Let S denote a strongly convex subgraph of a connected abelian homogeneous graph. Suppose a function f : V --+ lR satisfies the Dirichlet or Neu­ mann boundary condition and achieves the log-Sobolev constant o . Also, assume L f 2 (x)d:r vol S . Then the log-Sobolev constant o of G satisfies =

:r E S

>

0 -

mm ( .

AI

1

)

4 ' 24kD2 log U2 '

where U supz 1/(z) l , k is the degree, and D denotes the diameter of S, and denotes the first eigenvalue of the Laplacian of G. =

AI

A k-regular abelian homogenous graph or a strongly convex subgraph has the eigenvalue bound 1 AI � 8 k D 2 and this lower bound is sharp up to a constant factor (the factor of k is necessary for some homogeneous graphs) . As a consequence of Theorems 12.4 , 12.9, and 12. 10, the log-Sobolev constant and the eigenvalue A1 can differ by at most a factor of log U . For graphs with isoperimetric dimension 6 (defined in Section 1 1 . 1 ) , we can derive a lower bound for o in terms of 6. THEOREM 1 2 . 14 . Let S denote a strongly convex subgraph of a connected ho­ mogeneous invariant graph with isoperimetric dimension 6. Suppose a function f : V --+ lR satisfies the Dirichlet or Neumann boundary condition and achieves the log-Sobolev constant. Also, assume L f 2 (x)dz vol S. Then the log-Sobolev zE S constant o of G satisfies =

0

.

� mm (

1 AI ) 4 ' 24kD26 log 6 '

1 2 .5.

THE ISOPERI M ETRIC DIMENSION AND THE SOBOLEV INEQUALITY

where

191

U = sup lf(z) i ,

kofisthetheLaplacian degree, and D denotes the diameter of S, and A1 denotes the first eigenvalue of G . z

1 2 .5. The isoperimetric dimension and the Sobolev inequality

We recall that a graph G has isoperimetric dimension d with an isoperimetric constant C& if for every subset X of V(G) , the total sum of edge weights between X and the complement X of X satisfies Wr , y

·-·

� C& (vol X) --r

::; vol X and C& is a constant depending only on d. z E X,y�X

where vol X

We will establish the following convergence bound using the isoperimetric di­ mension. In a weighted graph G with isoperimetric dimension d and isoperimetric constant cc� , we have Ll (t) ::; e-c if t>

c

' c

- + -

- Ad

where d = log( c'' c;j 1 ) for some absolute constant d' . A

To prove this, we will need the following version of the Sobolev inequality (see Section 1 1 .3 and [60]) : For any function g : V(G) --+ IR and for d > 2,

L \g ( u) - g(v) i 2 w(x, y) � c i�f( L \g(v) - mi 2 "Ydv ) �

(12.30)

v

where 'Y = c! � 2 and c = c6 (o - 1 ) 2 /4d2 . We will prove a discrete version of the classical regularity theory of De-Georgi­ Nash-Moser [25 1]: THEOREM 1 2 . 1 5 . For a weighted graph G with isoperimetric dimension o, sup­ pose f is a harmonic eigenfunction with eigenvalue A . Then we have

sup I f2 ( x) "'

1 ::; cc! / 2

Ac!/2

L f 2 ( x) dr "'

for some absolute constant c . PROOF. By (12.13) , we have

L A i fP (x) id., r

=

=

( 12.31 )

L l f P - 1 (x) l L f(x)

L:U( x ) - f(y)) (JP- 1 (x) - r- 1 (y))w(x, y) ·

12.

19 2

ADVANCED TECHNIQUES FOR RANDOM WALKS ON GRAPHS

I = JP 2

Now we use the Sobolev inequality ( 12.30) with g

and also (12.31) . We have

z ,y

Or, for q :;::: 0, ( 12.32) We apply (12.32) recursively. For q

where c 1

= c- 1 .

=

(�f"(x)d, )"

2, we have ,;

'

c, 4 A

Setting q = 2')' we have

� f'(x)d..

Extending to q = 2')' i , we get

We now note that 1 1 1 + - + · · · + ----:')' '

/'

Moreover, 4

(�) ( 1h

2')' - 1

4 4 ')' 2')'2 - 1

)

<

1 1--

-1

sup I :r

f2(x) I�

{J

2

/'

1 h2 <

46/2 .! + :\ + · · · + ...!..

<

46/2 /'6/( 2 -r-2)

<

Therefore we have

=

41 2 >.6 12

L

/' �

( 4e 2 ) 6/2 .

j2 (x)d.,

�

�·

1 2 .5. where

c2

=

THE ISOPERIMETRIC DIMENSION AND THE SOBOLEV INEQUALITY

1 93

4e2c1 • The proof of Theorem 12.15 is complete.

0

We note that in the proof of Theorem 1 2 . 15, it suffices to start with a function f satisfying {12.31} . Here we state a slightly more general version of Theorem 12. 1 5 . COROLLARY 1 2 . 1 6 . For a weighted graph G with isoperimetric dimension a, suppose f is a function satisfying

L ..\fP (x)d.,

;:::

L JP -1 (x )

L f ( x) .

"

Then we have sup I "'

f2 (x)

��

cli/2 .,\ 6 /2 L P ( x ) d., "'

for some absolute constant c.

As a consequence of the above theorem, we have THEOREM 1 2 . 1 7 . In a weighted graph G with isoperimetric dimension a and

.

isoperimetric constant c6 , the random walk approaches stationanty in 0 (

I ) steps where c' = log(c" c,! 1 ) fo I

An- 1 - 1 and A =

some absolute constant c" and .,\ 2..\ t f(>.. l + An- d otherwise. r

P ROOF. We replace f by g., =

can be checked that

=

log vol G

+ ..\a At if 1 - ..\ 1 ;:::

'lj;.,1f-11 2 ( Ht - /0}1f-1 /2 in Theorem 12 .15.

It

vol G 1/Jz1r-lf 2 (H2 t - lo ) 1f- l l21j;z

L g� (y )d11 y

< <

e- 2 tA II7r- 1/21/Jx ll 2 vol G _ 2 t>. vol G -- . e

Since gz is orthogonal to the eigenfunction ¢>o of £, we have e- t� L g�(y)d11 ;::: L g� - 1 ( y) L gz (y ) .

d:x

y

y

By C orollary 12. 16, we have sup I

g� (x) � � cli f2 e- Mli /2 L g� (y )dy . y

Finally, we consider .:l( t)

max =

where

IPt (y, x) - 1r(x) l 1r ( x )

z,y

max max lg., (y ) l z

11

<

max clif2 e - tMf2 L g� ( y) dy

<

vol G c6 /2 e - t� 6f 2 minx . d;z

z

c = c"c,! 1 . Theorem 12. 1 7 is proved.

11

0

1 94

12.

ADVANCED TECHNIQUES FOR RANDOM WALKS ON G RAPHS

Many invariants of the hypercube Qn can be effectively evaluated using the fact that Qn is the cartesian product of an edge. The convergence bound for the random walk problem on Qn is of order n log n (as described at the end of Chapter 1}. When dealing with subgraphs of Qn , the random walk problems become more complicated. We say Q� is a punctured cube if it is a subgraph of Qn obtained by removing one vertex from Q n . We will use the isoperimetric dimension approach as follows: The isoperimetric problems on hypercubes have long been studied. For a fixed number m, the edge boundary of a subset X � V(Qn) with vol X = m, is minimized when X is close to a Hamming ball (i.e. , Nr (v) for some v and r ) . In other words, the isoperimetric dimension for Q n is 5 n / log n and the isoperimetric constant C6 = 1. Using Theorem 12. 17, we have log vol Qn � (t ) < 5 .A < n log n. For a punctured cube Q�, the isoperimetric dimension stays virtually unchanged. So it follows from Theorem 12. 17, that its (lazy) random walk converges in time O(n log n) . =

Bibliography [1] H . L . Abbott , Lower bounds for some Ramsey numbers, Discrete Math. 2 ( 1972 ) , 289-293. [2] A. Agresti, Categorical Data A nalysis, John Wiley and Sons, New York, 1990. [3] D . Aldous, On the Markov-chain simulation method for uniform combinatorial simulation and simulated annealing, Prob. Eng. Info . Sci. 1 ( 1987) , 33-46. [4] D . Aldous, Some inequalities for reversible Markov chains, J. London Math. Soc. 25 ( 1982), 564-576. [5] N. Alon , Eigenvalues and expanders, Combinatorica 6 ( 1 986) , 86-96. [6] N . Alon and F.R.K. Chung, Explicit constructions of linear-sized tolerant networks, Discrete Math. 72 ( 1 988) , 1 5-20. [7] N . Alon , F.R.K. Chung and R.L. Graham, Routing permutations on graphs via matchings, SIA M J. Disc. Math. 7 ( 1994 ) , 513-530. [8] N. Alon, Z. Galil and V . D . Milman, Better expanders and superconcentrators, J. A lgorithms 8 ( 1987) , 337-347. [9] N. Alon and V. D. Milman, A t isoperimetric inequalities for graphs and superconcentrators, J. Comb . Theory B 38 ( 1985), 73-88. [10] N. Alon, Z . Galil and 0. Margalit, On the exponent of the all pairs shortest paths prob­ lem, 32nd Symposium on Foundations of Computer Science, IEEE Computer Society Press, ( 1 99 1 ) , 569-575 . [ 1 1 ] N. Alon and N. Kahale, Approximating t h e independence number v i a t h e (} function , preprint. [12] N. Alon and J.H. Spencer, The Probabilistic Method, John Wiley and Sons, New York 199 1 . [13] S . Arora and S . Safra, Probabilistic checking o f proofs; a new characterization o f N P . 33nd Symposium on Foundations of Computer Science, IEEE Computer Society Press, ( 1992), 2- 13. [ 14] L . Babai and M . Szegedy, Local expansion of symmetrical graphs, Combinatorics, Probability and Computing 1 ( 1991 ) , 1 - 1 2. [15] L. Babai , Automorphism groups, isomorphism, reconstruction, Handbook of Combinatorics ( eds. R. L. Graham, Gr6tschel and L. Lovasz ) , North-Holland, Amsterdam, ( 1 996) , 14471 540. [16] D . Bakry, L'hypercontractivite et son utilisation en theorie des semigroups , In Ecole d ' e t e de Saint Fleur 1 gg2, Springer Lecture Notes 1581 . [ 1 7] L.A. Bass alygo, Asymptotically optimal switching circuits, Problems Inform. Transmission 17 ( 1981 ) , 206-21 1 . [18] J . Beck, O n size Ramsey number o f paths, trees and circuits 1 . , J . Groph Theory 7 ( 1 983 ) , 1 1 5- 129. [19] W. Beckner, Inequalities in Fourier analysis, Annals of Mathematics 1 0 2 ( 1 975) , 159- 182. [20] V.E. Ben�s, Mathematical Theory of Connecting Networks, Academic Press, New York 1965 . [21] J.C. Bermond and B. Bollobas, The diameter of graphs - a survey, Congressus Numerontium 32 ( 1981 ) , 3-27. [22] J .C. Bermond, C. Delorme and G. Farhi , Large graphs with given degree and diameter, I I I , Proc. Coli. Cambridge ( 1 98 1 ) , A nn. Discr. Math. 1 3 , North Holland, Amsterdam ( 1 982) , 23-32. [23] A. J . Berstein, Maximally connected arrays on the n-cube, SIA M J. Appl. Math. 15 ( 1 967) , 1485- 1 489. [24] F . Bien, Constructions of telephone networks by group representations, Notices A mer. Math. Soc. 36 ( 1989) , 5-22.

1 95

196

B IBLIOGRAPHY

[25] N.L. Biggs , A lg e b raic Graph Theory, (2 nd ed . ) , Cambridge University Press, Camb ridge, 1993. [26] N . L . B i ggs and M.H. H oare , The sextet construction for cubic graphs, Combinatorica 3 { 1 983) , 153- 165. [27] N.L. Biggs, E.K. Lloyd and R.J . Wilson, Graph Theory 1 736- 1 936, C l are n don Press, Oxford, 1976. (28] Y. Bishop, S . Fi e nberg, P. Holland, Discrete Multivariate A nalysis, MIT Press, Cambridge, 1975. [29] M . Blum, R.M . K arp, O.Vornberger, C. H . P ap ad imitri ou , and M . Yannakakis, The com plex ity of testing whether a graph is a superconcentrator, Inf. Proc. Letters 13 { 1981), 164- 167. [30] B. Bollobas, Random Graphs, Academic Press, New York ( 1 987) . [31] B. Bollobas, Extremal Graph Theory, Academic Press , London ( 1978). [32] B . Bollobas and F . R.K. Chung, The diameter of a cycle plus a random m atching, S/A M J. on Discrete Math e m a t ic s 1 ( 1 988) , 328-333. [33] B . Bollobas and I. Leade r , Edge- isoperimetric inequalities in the grid , Combinatorica 1 1 ( 1991) , 299-314. [34] B . Bollobas and I . Leader, An isoperimetric i nequality on the discrete torus, SIA M J. Disc. Math. 3 ( 1990), 32-37. [35] B. Bollobas, and A . Thomason, Graphs which contain all small gr aphs , European J. of Com­ binatorics 2 ( 1 98 1 ) , 13- 1 5 . [36] B. Bollobas and W.F. d e Ia Vega, The diameter o f random graph s , Combinatorica 2 { 1 982) , 125-134. [37] J.A. Bondy and M . Simonovits, Cycles of even l engt h i n graphs, J . Combin. Theory Ser. B 16 ( 1 974) , 97- 1 05. [38] R.B. Boppan a, E i genvalues and graph bisect ion: A n average-c ase an aly si s , 28nd Symposium on Foundations of Computer Science, IEEE Computer Society Press, ( 1987) , 280-285. [39] R. Bott and J . P. May be rry, Matrices and trees , In Economic A ctiv ity A naly s i s , (0. Morgen­ st e rn , ed ) , John Wiley and Sons, New York ( 1 954) , 39 1-340. (40] A. Broder, A. Frieze and E. Upfal, Existence and const ructio n of edge d isj oi nt p ath s on ex pand er graphs , Proc. Sym. Theo. on Compu t ing , ACM ( 1 992), 140- 1 49. [41] R. Brooks, The s p ect ral geometry of k- reg ular g raphs, Journal d'Analyse M athematique, 57 ( 199 1 ) , 120- 1 5 1 . [4 2] N . G . de B ru ij n , A combinatorial p robl em , Nederl. A k a d. Wetensch. Proc. 49 ( 1 946) , 758-764. [43] D . A . Burgess, On character sums and pri miti ve roots, Proc. London Math. Soc. 12 ( 1 962) 1 79- 192. (44] P. Buser, Cay ley g rap hs and planar isospectral domains, in Geome try a n d A nalysis on Man­ ifolds ( T. Sunada, ed . ) , Springer Lecture Notes 1339 ( 1988) , 64-77. (45] P. Buser, Cubic grap hs and the first e i genvalue of a Rie mann surfac e , Math. Z. 162 ( 1 978) , 87-99. [46] L . C ac ce t t a , O n ext remal graphs with given diameter and con n ec t i vity, A nn. New York Acad. Sci. 328 ( 1 979) , 76-94 . [47] A . Cayley, A theorem on trees , Quart. J. Ma th . 23 ( 1 889), 376-378. (48] J. C h eeger , A lower bound for the smallest eigenvalue of the L apl aci an, Problems i n A n alysis ( R. C. Gunning, ed . ) , Princeton Univ. Press ( 1 970) , 1 95- 199. [49] S i u Yuen C h en g , Peter Li a n d Shi n g- Tu ng Yau , On the upper esti m ate of the heat kernel of a c omp lete Riemannian man i fold , A m e rican Journal of Mathematics 103 ( 1 98 1 ) , 102 1 - 1 063. [50] R. C h rist e n se n , Log-Linear Models, Springer- Verlag, New Yor k , 1990. [5 1] F . R . K . C hu ng , Diameters and eigenvalues, J. of A mer. Math. Soc. 2 ( 1 989) , 1 87- 1 96. [52] F.R.K. Chu n g , Eigenvalu es of gr aphs and Cheeger inequalities, i n Combinatorics, Paul Erdos is Eighty, Volume 2, edited by D. M i k l 6s , V. T. S6s, and T. Szony i , J anos B o l y ai Mathematical Soc i e ty, B udapest ( 1 996 ) , 157- 1 72. [5 3] F . R. K . C h ung , V . Faber and T. A . M anteuffel , On t h e diameter of a graph from eigenvalues associated with its L ap lac i an , SIA M. J. Discrete Math. 7( 1 994) , 443-457. (54] F . R . K . Chung, A . G r igo r yan , and S.-T. Yau , Upper bounds for eigen values of the d i screte and conti nuous Laplace operators, A dvances in Mathematics 1 1 7 ( 1 996 ) , 165- 1 78 . [55] F . R . K . Chu n g , A . Grigor 'yan , and S .-T. Yau , Ei genval u es and diameters for manifolds an d grap hs , Tsing Hua Lectures on Geometry and A n alysis, to appear. [56] F . R . K . Ch u n g and S .-T. Yau , A H arnack inequ ality for D i richlet eigenval ues , preprin t . -

.

'

BIBLIOGRAPHY

197

[57] F . R. K . Chung and Prasad Tetali, Isoperimetric inequa.lities for cartesian products of graphs, preprint. [58] F.R.K. Chung and S .-T. Yau, A Harnack inequality for ho mogeneous graphs and s u bg raphs Communications in A nalysis and Geometry, 2 ( 1 994) , 628-639. (59] F . R. K . Chung and S .-T. Yau, Eigenvalues of graphs and Sobolev i nequalities, Combinatorics, Probability and Computing, 4 ( 1 995) , 1 1-26. (60] F.R.K. Chung and S .-T. Yau, Eigenvalue inequalities of graphs and convex subgraphs, Communications in A nalysis and Geometry, to appear. [61] F.R.K. Chung and S .-T. Yau, Eigenvalues, flows and separators o f graph s , preprint. (62] F . R . K . Chung and S .-T. Yau, Logarithmic Harnack i nequalities, preprint. (63] F . R . K . Chung, On concentrators, superconcentrators, generalizers and nonblocking networks, Bell Systems Tech. J. 58 ( 1 978), 1 765- 1777. (64] F . R. K . Chung, A note on constructive methods for Ram sey n umb ers , J. Graph Th. 5 ( 1 98 1 ) , 109- 1 13. (65] F . R. K . Chung, Diameters of communications networks, Mathematics of Information Pro­ cessing, AMS Short Course Lecture Notes ( 1 984) , 1 - 1 8 . (66] F.R.K. Chung, Diameters of graphs: O l d problems and new results, Congressus Numemntium 60 ( 1 987), 295-317. (67] F.R.K. Chung, Quasi-random classes of hypergraphs, Random Structures a n d A lgorithms 1 ( 1 990) , 363-382. (68] F . R . K . Chung and M.R. Garey, Diameter bounds for altered graphs, J. of Graph Theory 8 ( 1 984), 5 1 1-534. (69] F.R.K. Chung, Constructing random-like graphs, in Probabilistic Combinatorics and Its A p ­ plications, ( B . B o ll ob as e d . ) , Amer. Math. Soc . , P rovidence, ( 199 1 ) 21-56 (70] F . R . K . Chung and R.L. Graham, Qu asi ran dom hype rgraphs , Random Structures and A lgo­ rithms 1 { 1990 ) , 1 05- 124. (71] F.R.K. Chung and R.L. Graham, Q u asi random tournaments, J. o f Graph Theory 1 5 ( 1 991 ) , 1 73- 198. (72] F . R . K . Chung and R.L. G raham , Maximum cuts and q uas i r an d om graphs , Random Graphs (Alan Frieze and Tom asz Luczak, ed s. ) , John Wiley and Sons, New York ( 1 992) , 23-34 . (73] F . R . K . Chung and R.L. Graham, On graphs not containing prescribed induced subgraphs, in A Tribute to Paul Erdos, (A. Baker et al . eds.) Cambridge University Press ( 1 990) , 1 1 1- 1 20 . (74] F . R. K . Chu ng and R.L. Graham, Quasi-random set systems, J. A mer. Math. Soc. 4 ( 1991) , 151- 196. (75] F . R . K . Chung and R.L. G raham , Quasi-random subsets of Zn , J. Combin. Th. (A) 6 1 ( 1 992) , 64-86. (76] F . R . K . Chung, R . L. G raham and R.M. Wilso n , Quasi-random graphs, Combinat orica 9 { 1989) , 345-362. (77] F . R . K . Chung, The regularity lemma for hypergraphs and quasi-randomness, Random Struc­ tures and A lgorithms 2 ( 1991), 241-252. (78] F. R . K . Chung and R.L. Graham , Cohomological aspects of hypergraphs, Trans . A mer. Math. Soc. 334 ( 1 992) , 365-388 (79] F . R . K . Chung, R.L. Graham and S .-T. Yau , On sampling with Markov chains, Random Structures and A lgorithms 9 ( 1 996) 55-78. (80] F . R . K . Chung and R.L. Graham, Random walks on generating sets of groups, preprint . (8 1 ] F . R . K . Chung and R.L. G r ah am , Stratified random walks on an n-cube, preprint . (82] F. R . K . Chung and C . M . Grinstead, A survey of bounds for cla&�ical Ramsey numbers , J. Graph Theory 7 ( 1983) , 25-38. (83] F . R . K . Chung and K . Oden , Weighted graph Laplacians and isoperimetric inequalities , preprint . [84] F . R . K . Chung and S. Sternberg, Laplacian and vibrational spectra for homogenous gr ap hs, J. Graph Theory 1 6 ( 1 992) , 605-627. [85] F . R. K . Chung and S. Sternberg, Mathematics and the Buckyball, A merican Scientist, 81 ( 1 993), 56- 7 1 . (86] F . R . K . Chung, 8 . Kostant and S . Sternberg, Groups an d t h e Buckybal l , in L i e Theory and Geometry: In honor of Bertram Kostant ( Eds. J . - L . Brylinski, R. B ry l i ns k i V. Guillemin and V . Kac ) PM 1 23, Birkhauser, Boston, 1994. (87] F . R . K . Chung, D . Rockmore, S. Sternberg, On the symmetry of the Buckyball, preprint .

,

-

-

-

,

BIBLIOGRAPHY

1 98

(88] F . R.K. Chung and P. Tetali, Communication complexity and quasi-randomness, SIA M J. Discrete Math. 6 ( 1993) , 1 1 0-- 1 23. (89] F . R. K . Chung and R.P. Langlands, A combinatorial Laplacian with vertex weights, J . Comb. Theory (A), 75 ( 1 996) , 316-327. (90] D . M. Cvetkovic, M. Doob and H. Sachs, Spectra of Graphs, Theory and Application, Aca­ demic Press, 1980. (91] D . M . Cvetkovic, M . Doob, I . Gutman , and A. Torgaiiev , Recent results i n the Theory of Graph Spectra, North Holland , Amsterdam 1988. (92] E . B. Davies, Heat kernel bounds, conservation of probability and the Feller property, J. d 'A nalyse Math. 58, ( 1 992) , 99-1 19. (93] P.J . Davis, Circulant Matrices, John Wiley and Sons, New York ( 1 979) . (94] P . Deligne, L a conjecture d e Wei! I, Inst. Hautes Etudes Sci. Publ. Math 4 3 ( 1 974) 273-307. (95] E. D 'Hoker and D . H . Phong, On determinants of Laplacians on Riemann surfaces, Com m . Math. Phys. 1 0 4 ( 1 986) , 537-545. (96] J . - D Deuschel and D.W. Stroock, Hypercontractivity and spectral gap of symmetric diffu­ sions with applications to the stochastic Ising models, J. Funct. A nal. 92 ( 1990 ) , 30-48. (97] P. Diaconis, Group Representations in Probability and Statistics, Institute of M ath. Statistics, Hayward, California, 1988. (98] P. Diaconis and D.W. Stroock, Geometric bounds for eigenvalues of M arkov chains, A nnals Applied Prob. 1 ( 1991 ) , 36- 6 1 . (99] P . Diaconis and L. Saloff-Coste, Comparison theorems for reversible Markov chains, A nnals of A pplied Prob. 3 ( 1993) , 696-730. [100] P. Diaconis and L . Saloff-Coste, Comparison techniques for random walks on finite groups A nnals of Applied Prob . 4 ( 1993) , 2 1 3 1-2156. [ 1 0 1 ] P. Diaconis and L. Saloff- Coste, Logarithmic Sobolev inequalities for finite M arkov chains, preprint. [ 1 02] P. Diaconis and L. Saloff-Coste, Walks on generating sets of groups, preprint. [ 1 03] P. Diaconis and B . Sturmfels, Algebraic algorithms for sampling from conditional distribu­ tions, preprint. [ 1 04] P. Diaconis, R.L. Graham and J. Morrison, Asymptotic analysis of a random walk on a hypercube with many dimensions, Random Structures and Algorithms 1 ( 1 990), 5 1-72. [105] P. Diaconis and L . Saloff-Coste, An application of Harnack inequalities to random walk on nilpotent quotients, preprint. [ 1 06] M . J . Dinneen , M . R. Fellows and V. Faber, Algebraic constructions of efficient broadcast networks, Lecture Notes in Comput. Sci. , 539, Springer, Berlin, 199 1 , 152- 158. [ 1 07] J. Dodziuk and L. Karp, Spectral and function theory for combinatorial Laplacians, in Geometry of Random Motion, Contemp. Math 73, AMS Publication ( 1 988) , 25-40. [ 1 08] M. Dyer, A. Frieze and R. Kannan , A random polynomial time algorithm for approximating the volume of convex bodies, JA CM 38 (1991), 1-17. [109] M . Eichler, Quaternary quadratic forms and t he Riemann hypothesis for congruence zeta functions, A rch. Math. 5 ( 1 954) , 355-366. [ 1 10] B. Elspas, Topological constraints on interconnection limited logic, Switching Circuit Theory and Logical Design 5 ( 1 964) , 1 33- 147. [ 1 1 1] P. Erdi5s , Some remarks on the theory of graphs, Bull. A mer. Math. S oc . 53 ( 1947 ) , 292-294. [ 1 12] P. Erdos, Some remarks on chromatic graphs, Colloquium Mathematicum 1 6 ( 1967) , 253256 . ( 1 13] P . Erdi5s, S . Fajtlowicz an d A.J. Hoffman , Maximum degree in graphs o f diameter 2, Net­ works 1 0 ( 1 980) , 87-90. ( 1 14] P. Erdos and A . Hajnal , On spanned subgraphs of graphs, Betrage zur Graphentheorie und deren A nwendungen, Kolloq. Oberhof (DDR) ( 1977) , 80-96. ( 1 1 5] P. Erdi5s and A . Renyi , On a problem in the theory of graphs, Publ. Math. Inst. Hungar. A cad. Sci. 7 ( 1 962) , 623-64 1 . ( 1 16] P . Erdos, A . Renyi and V.T. S6s , O n a problem of graph theory, Studia Sci. Math. Hungar. 1 ( 1966} , 21 5-235. [ 1 17] P. Erdos and H . Sachs, Regulare Graphen gegenebener Teillenweite mit Minimaler Knoten­ zahl, Wiss. Z. Univ. Halle - Wittenberg, Math. Nat. R . 12 ( 1 963) , 251-258. ( 1 18] P. Erdos and V.T. S6s, On Ramsey-Turan type theorems for hypergraphs, Combinatorica 2 ( 1 982) , 289-295. .

BIBLIOGRAPHY

1 99

[ 1 19] P. Erdos and J . Spencer, Imbalances in k-colorations, Networks 1 { 1 972) , 379-386. [ 1 20] P. Erdos and J. Spencer, Probabilistic Methods in Com b in a tori cs, Academic Press , New York { 1974 ) . [ 1 2 1 ] R.J. Faudree and M . Simonovits, On a class of degenerate extremal graph problems, Com­ binatorica 3 { 1 983) , 97- 107. [ 1 22] U. Feige, Randomized graph products, chromatic numbers, and the Lovasz 9-function, Proc . Sym. Theo. on Computing, ACM { 1995), 635-640. [ 1 23] U. Feige, S. Goldwasse r, L. Lovasz, S. Safra, and M. Szegedy, Approximating clique is al­ most N ?-complete, 32nd Symposium on Foundations of Computer Science, IEEE Computer Society Press, { 1991 ) , 2- 1 2 . [ 1 24] P. Frankl and V. R&ll , Forbidden intersections, 'Ihlns. A MS 300 { 1987) , 259-286. [ 125] J . A . Fill, Eigenvalue bounds on convergence to stationarity for nonreversible Markov chains, with an application to the exclusion process, Ann. Appl. P ro b . 1 { 1 991) 62-87. [ 1 26] M. Friedler, Algebraic connectivity of graphs , Czech. Math. J. 23 {98) { 1 973) , 298-305. [1 27] L.R. Ford and D.R. Fulkerson, Flows i n Netwo rk s , Princeton Univ. Press { 1 962) . [ 1 28] P . Frank l , A constructive lower bound for some Ramsey numbers, A rs Combinatoria 3 { 1 977) , 297-302. [ 1 29] P. Frankl, V. R&ll and R . M . Wilson , The number of submatrices of given type in a Hadamard matrix and related results, J. Co mb in ato ri al Th. (B) 44 { 1988) , 31 7-328. [ 130] P. Frankl and R . M . Wilson, Intersection theorems with geometric con seq uences , Combina­ torica 1 ( 198 1 ) , 357-368. [131] M . L . Fredman , New bounds on the complex ity of the sho rt es t path proble m , SIA M J. Computing 5 { 1 976) , 83-89. [132] Joel Friedman , On the second eigenvalue and random walks in random d-regular graphs, Combinatorica 11 { 1 99 1 ) , 331-362. [ 1 33] J. Friedman and N . Pippenger, Expanding graphs contain all small trees, Com binatorica 7 ( 1987) , 71-76 . [ 1 34] G . Frobenius, Uber d i e Charaktere der alternierenden Gruppe , Sitzungsberichte Preuss. A kad. Wiss. { 190 1 ) , 303-315. [ 1 35] G . Frobenius, Uber Matrizen aus nicht negative Elementen, Sitzber. Akad. Wis s . Berlin { 19 1 2 ) , 456-477. [ 1 36] Z. Fiiredi and J. Koml6s, The eigenvalues of random symmetric matrices, Combinatorica 1 ( 198 1 ) , 233-24 1 . [1 37] 0. G abber and Z. Galil, E x p li cit construction o f linear sized superconcentrators , J . Co mp u t . Syst e m Sci. 22 { 198 1 ) , 407-420. [ 138] A. Ga.ltma.n , La.pla.cia.ns , the Lovasz number, and Delsa.rte's linear programm ing bound, preprint. [ 1 39] F.R. Gantmacher, The Theory of Matrices, Vol . 1, Chelsea. Pub. Co. , New York ( 1977 ) . [ 1 40] M . R. Garey and D . S . Johnson , Compu t e rs a n d In tractability, A Guide to the Theory of NP- Completeness, W. H . Freeman and Co. , San Francisco, 1979. [ 1 4 1 ] M. Goem a.ns and D . Williamson , Improved approximation algor i t hm s for maximum cut and sa.tisfia.bility problems using semidefinite programming, J. ACM 42 ( 1 995), 1 1 1 5- 1 145. [ 1 42] P. G i n s ba.rg , A pplied Conformal Fi el d Theory, l e s Houches { 1 988 ) . [ 143] W. Goddard, private communi c at ion. [144] R. L . Graham and V. Rod I , Nu m be rs in Ramsey theory, Surveys in Com binatorics ( ! 9 8 7 ) , { C. Whitehead , ed . ) London Math. Soc . Lecture Notes Series 1 2 3 , 1 1 1 - 1 53. ( 145] S . W . Graham and C . Ringrose, Lower bounds for least quadratic n on- re sid u es , A nalytic Number Theory, ( e d it ed by B. B ern dt , H. Halberstam, H . D ia mo nd and A. Hi ldebrand ) , Birkhauser, Boston 1990. [ 1 46] R.L. G raham and J . H . Spencer , A constructi ve solution to a. tournament problem, Ca nad. Math. Bull. 1 4 { 1 971 ) , 45-48. [ 1 4 7] A . Grigor'ya.n , I ntegral m ax imum principle and its app li cations, Proc. of E di nb u rg h Ro yal Society, to appear. [148] M. Gromov, Groups of po l yno mial growth and expanding m aps , Publ. IHES, 53 { 1 98 1 ) , 53-78 . (1 49] L . Gross , Logarithmic Sobolev inequalit ies, A mer. J. Math. 91 ( 1 976 ) , 1 06 1 - 1 083. [1 50] L. G ross, Logarithmic Sobolev i n equalities and controctivity properties of semi-groups, Springer Lecture Notes 1 563. { 1 993 ) .

200

BIBLIOGRAPHY

[ 1 5 1 ] M. G rotschel , L. Lovasz and A . Schrijver, The ellipsoid method and its consequences in combinatorial optimization, Combinatorica 1 ( 198 1 ) , 169- 197. [ 1 52] M . G rotschel, L. Lovasz and A . Schrijver, Geometric Algorithms and Combinatorial Opti­ mization, Springer-Verlag, Berlin, 1 988. [ 1 53] W. Haemers, Eigenvalue methods in Packing and Covering in Combinatorics, ( A . Schrijver ed . ) , Mathematisch Centrum , Amsterdam ( 1 982) , 15-38. [ 154] M . M . Halldorsso n, A still better performance quarantee for approximating graph coloring, Information Processing Letters 45 ( 1 993), 19-23. [ 1 55] L. H. Harper, Optimal numberings and isoperimetric problems on graphs, J. of Com b . Theory 1 ( 1 966), 385-393. [ 1 56] S . Hart , A note on the edges of the n-cube, Discrete Math. 1 4 ( 1 976) , 157- 1 63. [ 157] J. Hastad, Clique is hard to approximate within n 1 - • , preprint. [158] J. Haviland and A. Thomason, Pseudo-random hypergraphs, Discrete Math. 75 { 1989) , 255-278. [ 1 59] W. Hebisch and L. Saloff-Coste, Gaussian estimates for Markov chains and random walks on groups, preprint . [ 1 60] A . J . Hoffman and R.R. Singleton , On Moore graphs with diameter 2 and 3, IBM J. of Res. Development 4 ( 1 960) , 497-504 . ( 1 6 1 ] A. J. Hoffm an , Eigenvalues of graphs, in Studies in Graph Theory II ( D . R. Fulkerson, ed . ) , M . A . A . Studies in Math, Washington D.C. , ( 1 975) , 225-245. ( 1 62] R. A. Horn and C . R. Johnson, Matrix A nalysis, Cambridge University Press, New York, 1985. ( 1 63] J . Igusa, Fibre systems of Jacobian varieties III, A merican J. of Math. 81 ( 1 959) , 453-476. ( 1 64] W. lmrich , Explicit construction of regular graphs without small cycles, Combinatorica 4 ( 1984 ) , 53-59. ( 1 65] K. Ireland and M. Rosen , A Classical Introduction to Modern Number Theory, Springer­ Verlag, New York 1982. (1 66] M . Jerrum and A . J. Sinclair, Approximating the permanent , SIA M J. Computing 1 8 ( 1989 ) , 1 1 49- 1 1 78 . ( 1 67] F . Juhasz, On t h e spectrum of a random graph , Colloq. Math. Soc. Janos Bolyai 2 5 , A lge­ braic Methods in Graphs Theory , Szeged ( 1978) , 3 1 3-316. ( 1 68] N . K ahale, Isoperimetric inequalities and eigenvalues, SIA M J. Discret e Math . , to appear. ( 1 69] B . S . Kashin and S.V. Konyagin, On systems of vectors in a Hilbert space, Trudy Mat. lnst. imeni V. A. Steklova 1 5 7 ( 198 1 ) 64-67. English translation in Proceedings of the Steklow Institute of Math. AMS ( 1 983), 67-70. (1 70) N .M. Katz, An estimate for character sums, J. A mer. Math. Soc. 2 { 1 989) , 1 97-200. ( 1 71] F . Kirchhoff, Uber die Auflosung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Strome gefiihrt wird , Ann. Phys. chem. 72 ( 1 847) , 497-508 (1 72] M. Klawe, Non-existence of one-dimensional expanding graphs, 22nd Symposium on Foun­ dations of Computer Science, IEEE Computer Society Press, { 1 98 1 ) , 109- 1 1 3. ( 1 73] D. K arger, R. M otwani, and M . Sudan, Approximate graph coloring by semi-definite pro­ gramming, 35th Symposium on Foundations of Computer science, IEEE Computer Society Press, 1994 , 2- 1 3 . (1 74] D . E . Knuth , Th e A rt of Computer Programming, vol . 3, p. 24 1 , Addison-Wesley, Reading, MA, 1 973. ( 1 75] D.E. Knut h , The sandwich theorem , Electronic J. of Combinatorics 1 ( 1994 ) , A l . ( 1 76] Robert P . Langlands and Yvan Saint-Aubin, Algebra-geometric aspects o f the Bethe equa­ tions, preprint. ( 1 77] F. Lazebnik, V . A. Ustimenko and A . J . Woldar, A new series of dense graphs of high girth, Bull. A mer. Math. S oc . 32 ( 1 995) , 73-79. ( 1 78) F.T. Leighton, Introduction to Parallel A lgorithms and A rchitectures: A rrays, Tree s , Hy­ percubes, Morgan-Kauffman , San Mateo, CA, 1992. (1 79] F.T. Leighton and Satish Rao, An approximate max-flow min-cut theorem for uniform mul­ ticommodity flow problem with applications to approximation algorithms, 29nd Symposium on Foundations of Computer Science, IEEE Computer Society Press, ( 1 988) , 422-431 . ( 1 80] T . Lengauer and R.E. Tarjan , Asymptotically tight bounds o n time-space trade-offs i n a pebble game, J. A ssoc. Compu t . Mach. 29 { 1982 ) , 1 087- 1 1 30. ( 1 8 1 ] H . Lenstra, personal communication .

BIBLIOGRAPHY

201

[1 82) G . Lev, Size bounds and parallel algorithms for networks, Ph.D. Thesis, Department of Computer Science, Uni versity of Edinburg . ( 1 83] W. Li, Character sums and abelian Ramanujan graphs, J. Number Theory 41 { 1 992) , 199217. [ 1 84) P. Li and S . T . Yau, On the parabolic kernel of the S chrodinger operator, A c t a Mathematica 156 ( 1 986) , 153 201 . (185) P. Li and S. T. Yau, Estimates of eigenvalues of a compact Riemannian manifold, A mer. Math. Soc. Proc. Symp. Pure Math. 36 ( 1 980) , 205-240. (186) J . H . Lindsey, Assignment of numbers to vert ices, Amer. Math. Monthly 71 { 1 964) , 508-516. (187) R.J. Lipton and R.E. Tarjan , A separator theorem for planar graphs , SIA M J. A ppl. Math. 36 ( 1 979) , 1 77- 189. (188) L . Lov as z , On the Shannon capacity of a graph 'Ihmsactions on Information Theory -IT25, IEEE Co m p u ter Society Press, ( 1 979) , 1-7. ( 1 89) L. Lovasz, Perfect graphs , i n Selected Topics in Graph Theory 2, { eds R . L. Wilson and L.W. Beineke) , Academic Press, New York { 1983 ) , 55-87. ( 1 90) L. Lovasz and M. Simonov its, Random walks in a con vex body and an i mp roved volume algorithm, Random Structures and A lgorithms 4 { 1 993 ) , 359-4 12. (191) A . Lubotzky, R. Phillips and P. Sarnak, Ramanujan graphs, Combinatorica 8 { 1 988), 261278. (192) G.A. Margulis, Explicit constructions of con cen trato rs , Problemy Peredaci Informacii 9 { 1 973) , 71-80 {English trans!. in Problems Inform. Transmission 9 { 1 975) , 325-332. ( 1 93) G.A. Marg ulis Arithmetic groups and graphs without short cycles , 6th In ternal. Symp. on Information Theory, Tashkent { 1 984) A bstracts 1, 123-125 (in Russian ) . [194) G . A . Margulis, Some new constructions o f low-d e n s ity parity check cod es , 3 rd In tern a l . -

,

,

{ 1 987) , 275-279 ( in Russian ) . [1 95) G . A . Mar gulis, Explicit group t h e oret ic const ruct ions of combinatorial schemes and th ei r ap­ plications for the construction of ex panders and concentrators , Problemy Peredaci Informacii { 1 988) (in Russian) . [ 1 96] J . C . Maxwell, A Treatise o n Elec tricity and Magne tism I, Oxford , Clarendon P ress { 1 892) , 403 4 1 0 . [ 197) B. Mohar, Isoperimetric numbe r of grap h s, J. of Comb. Theory (B) 47 { 1 989) , 274-29 1 . (198] R . J . McEliece, E . R . Rodemich , and H.C. Rumsey, J r . , The Lovasz and some ge n e ral i z at io n s, J. Combinatorics, Inform. Syst . Sci. 3 {1978) , 134- 152. (1 99) H .L. Montgo m ery, Topics in Multiplicative Number Theory, Lect ure Notes i n Mat h . 2 2 7 , Spring e r-Ve rlag , New York { 1 971 ) . (200) J . W. Moon, Counting Labelled Trees, Canadian M at h e matical Monographs, Canad i an Mathematical Congress, Montreal , 1970 . [20 1) Zs. Nagy, A co n st ructi ve est imation of the Ramsey numbers, Mat. Lapok 23 { 1 975) , 30 1 - 302. [202] A. Nilli , On the second eigenvalue of a gr ap h Discrete Math. 91 { 1 99 1 ) , 207- 2 1 0 . (203) B. Osgo o d , R. P h i l l ips , and P . Sarnak, Extremals o f determinants o f L ap l ac i ans , J. Funct. A nal. 80 { 1 988) , 1 48-2 1 1 . (204 ) B . Osgood , R . Phillips, and P . Sarnak, Moduli sp ac e, h e i gh ts and isospect ral se ts of plan e domai ns, A n n . Math. 129 ( 1989), 293-362. (205] E.M. Palmer, Graphical Evolution, Jo h n Wiley and Sons , New York ( 1 985) . ( 206) W.J. Paul, R .E. Tarjan and J . R. Celoni , Space bounds for a game on graphs , Math. Soc. Theory 10 { 1 977) , 239-25 1 . (207] 0 . Perron, Zur Theorie der Matrizen , Math. A n n . 6 4 ( 1 907 ) , 248-263. (208) M. Pinsker, On the complexity of a concentrator, 7th Internat . Teletraffic Conf. , Stockholm, June 1973, 31 8/ 1-318/4. (209] N . Pippenge r , S u perconcentrators, SIA M J. Comput. 6 ( 1 977) , 298 - 304. [2 10] N. Pippenger, Advances in pebbling, Internal. Collo . On A utomation Languages and Pro­ gramming 9 { 1 982 ) , 407-4 1 7. (2 l l ] G. Polya and S. Szego, Isoperimetric Inequ ali t ies in Mathematical Physics, A nnals of Math. Studies, no. 27, Princeton University Press, ( 1 95 1 ) . (2 12] A . M . Polyakov Quantum geometry of bosonic strings, Phys . Letters B 1 03 ( 1 98 1 ) , 207- 2 1 0 . (2 1 3] P. S arnak, Some A pplications of Modular Forms, C a mbr i dg e University P ress , Cambridge, 1990.

Seminar on Information Theory, convolution codes and multi-user communical.ion, Sochi

-

,

,

202

BIBLIOGRAPHY

[214] S . Ramanujan, On certain arithmetical functions, TI-ans. Cambridge Philos. Soc. 22 (9) ( 1 916), 159-184. [215] F.P. Ramsey, On a problem of formal logic, Proc. London Math. Soc . 30 ( 1930 ) , 264-286. [21 6] D. Ray and I.M. Singer, Analytic torsion for complex manifolds, Ann. Math. 98 ( 1 973) , 1 54. [217] D.K. Ray-Chaudihuri and R.M . Wilson, On !-designs, Osaka J. Math. 1 2 ( 1975 ) , 745-744. [21 8] A. Renyi , On the enumeration of trees, in Combinatorial Structures and their Applications ( R. Guy, H . Hanani, N . Sauer, and J. Schonheim, eds. ) Gordon and Breach, New York ( 1970 ) , 355-360. [219] V. ROd! , On the universality of graphs with uniformly distributed edges, Discrete Math. 59 ( 1 986) , 125- 134. [220] P. Sarnak, Determinants of Laplacians, Comm. Math. Phys. 110 ( 1987) , 1 13- 120. [221] A . Schrijver, A comparison of the Delsarte and Lovasz bounds, IEEE TI-ansactions on Information Theory IT-25, ( 1979) ,425-429. [222] J . J . Seidel, Graphs and their spectra, Combinatorics and Graph Theory, PWN-Polish Sci­ entific Publishers, Warsaw ( 1 989) , 147- 162. [223] R. Seidel , On the all-pairs-shortest-path problem , Proc. Sym. Theo. on Computing, ACM ( 1 992) , 745-749. [224] M. S imonovits and V.T. S6s, Szemen)di partitions and quasi-randomness, Random Struc­ tures and A lgorithms 2 ( 1 991 ) , 1 - 1 0 . [225] A.J. Sinclair, A lgorithms for Random Generation and Counting, Birkhauser, Boston, 1993. [226 ] A.J . Sinclair and M . R. Jerrum, Approximate counting, uniform generation, and rapidly mixing markov chains, Information and Computation 82 ( 1989) , 93- 133. [227] R. S ingleton, On minimal graphs of maximum even girth, J. Combin. Theory 1 ( 1966 ) , 306-332. [228] J. Spencer , Optimal ranking of tournaments, Networks 1 ( 197 1 ) , 135- 138. [229] J . Spencer, Ramsey's theorem - A new lower bound, J. Combinatorial Theory 1 8 ( 1975 ) , 108- 1 1 5 . [230] S . Sternberg, Group Theory i n Physics, Cambridge University Press ( 1992). [23 1] D.W. Stroock , Logarithmic Sobolev inequalities for Gibbs states, Springer Lecture Notes 1563, Berlin , ( 1 993) . [232] J . J . Sylvester, On the change of systems of independent variables, Quarterly Journal of Mathematics 1 ( 1 857) , 42-56. Collected Mathematical Papers, Cambridge 2 ( 1908 ) , 65-85. [233] R.M. Tanner, Explicit construction of concentrators from generalized N-gons, SIA M J. A l­ gebraic Discrete Methods 5 ( 1 984) , 287-294. [234] A . Thomason , Random graphs, strongly regular graphs and pseudo-random graphs, in Sur­ vey in Combinatorics ( 1 987) ( C. Whitehead , ed . ) , London Math. Soc. [235] A. Thomason , Pseudo-random graphs, Proc. Random Graphs , Poznan ( 1985) ( M . Kar6nski, ed . ) , Annals of Discrete Math. 33 ( 1 987), 307-33 1 . [236] A. Thomason, Dense expanders and pseudo-random bipartite graphs, Discrete Math. 75 ( 1 989 ) , 38 1-386. [237] M. Tompa, Time space tradeoffs for computing using connectivity properties of the circuits, J. Comp u t . System Sci. 1 0 ( 1 980 ) , 1 18- 132. [238] H . M . Trent , Note on the enumeration and listing of all possible trees in a connected linear graph , Proceedings of the National A cademy of Sciences, U. S. A . 40 ( 1 954), 1 004- 1007. [239] W.T. Tutte, Graph Theory, Addison Wesley, Reading, MA 1984. [240] G . E . Uhlenbeck and G.W. Ford, Lectures in Statistical Mechanics, Providence, American Mathematical Society ( 1 963) [24 1] L. G . Valiant , The complexity of computing the permanent , Theoret. Comput. Sci. 8 ( 1979 ) , 1 89-20 1 [242] L . G . Valiant , Graph theoretic properties in computational complexity, J. Comput. System Sci. 1 3 ( 1 976) , 278-285. [243] L . G . Valiant, A scheme for fast parallel communication , SIA M J. Comput. 11 ( 1 982) , 350361 . [ 2 44] N .T. Varopoulos, lsoperimetric inequalities and Markov chains, J. Funct. A nal. 63 ( 1985 ) , 2 1 5-239. [245] K. Vij ayan and U . S . R. M urty, On accessibility in graphs, Sankhya Ser. A. 26 ( 1964 ) , 299302. .

BIBLIOGRAPHY

2 03

[246) D.-L. Wang and P. Wang, Discrete isoperimetric problems, SIA M J. Appl. Math. 32 ( 1977) , 860-870. [247) A. Wei!, Sur les courbes algebrique et les varietes qui s'en deduisent, Actualites Sci. Ind. No. 1 0 4 1 ( 1 948) . [248) H. S . Wilf, The eigenvalues of a graph and its chromatic number, J. London Math. Soc. 42 ( 1 967) , 33o-332. [249) R. M. Wilson, Constructions and uses of pairwise balanced designs in Combinatorics ( M . Hall, Jr. , and J . H . van Lint , eds . ) Math Centre Tracts 5 5 , Amsterdam ( 1974), 18-4 1 . [250) A . Weiss, Girths of bipartite sextet graphs, Combina torica 4 ( 1984 ) , 241-245. [25 1) S. T. Yau and Richard M . Schoen , Differential Geometry, Science Publications, Beijing, 1988 (Chinese version} ; International Press, Cambridge, Mass achusetts, 1994 (English version) .

( 1r ; p ) -norm, 1 8 1 £2-norm, 1 5 x-squared distance, 18 k-access graph, 98 n-cube, 1 9 , 36 p-norm, 40

Index Chebyshev polynomial, 42 clique, 77, 87, 105 clique number, 1 05 co-NP-complete, 90 coboundary operator, 3 comparison theorems, 64 complete graph , 5 concentrator, 89 conductance, 90 connected , 6, 7, 24 consistent , 35 contingency table problem , 1 56 contraction , 12 convex subgraph, 137, 150 coset graph, 94 cycle, 6

abelian group, 136 adjacency matrix, 3 adjacent , 3 all distances algorithm, 42 all shortest paths algorithm, 42 almost regular, 78 aperiodicity, 14 automorphism, 1 9 automorphism group, 1 10

degree, 2 deviation , 78 diameter, 7, 4 1 , 102 diameter algorithm, 42 diameter-eigenvalue inequalities, 42 Dirichlet boundary condition , 123, 147, 190 Dirichlet eigenvalues, 123, 128-130, 132, 1 50 Dirichlet sum, 4 discrepancy, 7 1 , 74, 82 distance, 7, 43 distance transitive , 109, 1 1 4 , 1 1 6

bipartite, 6, 7 bipartite expander graph, 92 bipartite graph, 92 bond , 1 1 8 boundary conditions, 175 boundary expansion property, 137 boundary operator, 3 Buckybal l , 1 18 Buckyball graph, 1 09 cartesian product , 36, 62 Cayley graphs, 1 10 chain, 3 Chebyshev polynomial , 44 Cheeger constant, 24, 25, 56, 89, I l l characterization, 32 modified , 34, 35 weighted graphs, 35 Cheeger inequality, 26 chromatic number, 103

edge boundary, 1 23 edge expansion , 25 edge generating set , 1 1 0, 1 5 1 edge transitive, 1 1 2 edge boundary, 24 edge-cut, 23 edge-transitive, 1 1 0 eigenfunction , 4

205

206

INDEX

harmonic, 4

neighborhood, 70, 89, 90

Neumann bound ary condition ,

eigenvalue, 4 ergodic,

13

non-blocking network, 98 norma.!, 87

expander coefficient , 89 expander graph, 87-89

NP,

expansion factor, 89

NP-complete, 55

explicit construction , 87, 93

108

odd-even transposition sort, 61

flow, 55

orthonormal labeling, 106

Fo u rier c oefficients , 43

pairing, 63

girth,

Paley graph, 77, 93

103

Paley

graph , 2 group representation

theory,

117

135,

138

heat equ at i on , 1 45

heat kerne l , 145

heat kernel eigenvalue i nequ ality, 148 1 19

perfect graph, 105

grap h ,

117

Poincare inequalities, 6 4

p ol yn omial growt h rate , probabilistic m et h od , 87

group,

Ramanujan graph, 4 2 , 9 6 , 1 03

subgraph, 123

Ramsey property, 76

inner product , 4

random graph , 107

intersection grap h , 1 1 7

1 3 , 58 product of gr ap hs ,

random walk,

i nvariant , 136

cartesian

irreducibility, 1 4 isoperimetric di mension, 163, 1 79

i rreducible , 14 modified,

isoperimetric number, 34 , 90

Neuman n , 126

isotropy group, 1 1 0

ope rato r ,

4

determinant of , 1 3 1

Rayleigh quotient , 4 regular, 3

rel at i ve

po i ntwise distance , 1 6 , 1 6 1 , 1 77

Riemannian manifold, 4 rooted spanning forest , 1 2 9 , 1 32

graphs, 151 w al k , 1 5 , 1 7 , 1 77

lattice

route covering number, 63

log-Sobolev constant , 177, 1 83 with D i richlet boundary condition, 1 79 Neumann boundary condition , 1 79

logarithmic Harnack inequalities, 1 78 logarithmic Sobolev inequalities, 2 1 , 180

2

major access n etwork ,

15

reversible, 1 4

Laplace operator, 4 8 , 50, 1 5 3

98

route

set , 55

routing,

55

complete bipartite graph,

complete graph , 6 1 hypercube, 6 1 , 63 t ree, 6 1 routing assi g n m ent , 60

routing number, 60

Margulis graph , 95

separator, 23

matching, 99, 102 matrix multiplicat i o n , 42 matrix-tree theorem , 1 29

size , 23

measure, 4, 24

spanni n g tree, 1 29

moderate growth

37

ergodic, 13

isoperimetric ineq u ali t ies, 1 65

loop,

97

Raman spectrum , 1 2 1

independent set , 77

with

165

projection, 1 45

quatemion

hypercube, 36, 195

l az y

131

6

quasi-random , 69

Hooke 's law , 1 20

Laplace-Beltrami Lap l ac ian , 2

94

Plancherel formula, 1 9

Hamming balls , 36

h omogenous graphs,

path,

Petersen

Hamiltonian pat h , 55 Harnack inequ al ity,

sum graph,

partition function ,

weighted undirec t ed , 1 1

induced

1 2 3 , 146 , 190

Neumann eigenvalues, 123, 126, 1 50

entropy fu nct ion , 1 0 1

rate,

Moore bound , 1 0 1

1 65

Sobolev constant , 1 6 3 , 1 64 Sobolev inequalities, 1 63

spectrum , star, 6

4

61

INDEX stationary distribution , 1 3 strongly convex subgraph, 136, 1 90 support , 48 symmetric group, 60 total variation distance, 18 trace, 9 transpose, 3, 1 7 transposition, 60 vertex boundary, 24 , 123 vertex expansion, 24 vertex transitive, 109, 1 1 0 , 1 1 3 vertex weight, 4 vertex-cut, 23 vertex-transitive, 1 1 0 vertex-transitive graph, 1 9 vibrational Laplacian, 1 1 9 vibrational spectrum, 1 1 9 volume, 5 Voronoi region, 1 54 walk, 1 3 weight function, 35 weighted cartesian product, 180 weighted graph, 1 6 1 , 175

207