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.. · )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,!)
<
(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 ,
�)
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
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(
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 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
<
.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
�
_
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
<
.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)
=
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
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)
<
__ > 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
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 '
.. ) = 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
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
. ( 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
. L:l! 2(x) - f2(bx)] =
(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
.f2 (x )
L max 9 ¢19 (x)
bEW'
tPa (z)
. f2 (x ) 1 1 = (- f(x) + - "' L..J f(bx) - -f(gx) - -
=
wa .
w2
w2
wa
W
; 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
=
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)
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)
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
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)
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 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.
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
� (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
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
/'; 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)
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
)
