Isoperimetric problems

Transcription

1 CHAPTER 2 Isoperimetric problems 2.1. History One of the earliest problems in geometry was the isoperimetric problem, which was considered by the ancient Greeks. The problem is to find, among all closed cures of a gien length, the one which encloses the maimum area. The basic isoperimetric problem for graphs is essentially the same. Namely, remoe as little of the graph as possible to separate out a subset of ertices of some desired size. Here the size of a subset of ertices may mean the number of ertices, the number of edges, or some other appropriate measure defined on graphs. A typical case is to remoe as few edges as possible to disconnect the graph into two parts of almost equal size. Such problems are usually called separator problems and are particularly useful in a number of areas including recursie algorithms, network design, and parallel architectures for computers, for eample [183]. In a graph, a subset of edges which disconnects the graph is called a cut. Cuts arise naturally in the study of connectiity of graphs where the sizes of the disconnected parts are not of concern. Isoperimetric problems eamine optimal relations between the size of the cut and the sizes of the separated parts. Many different names are used for arious ersions of isoperimetric problems (such as the conductance of a graph, the isoperimetric number, etc.). The concepts are all quite similar, but the differences are due to the arying definitions of cuts and sizes. We will consider two types of cuts. A erte-cut is a subset of ertices whose remoal disconnects the graph. Similarly, an edge-cut is a subset of edges whose remoal separates the graph. The size of a subset of ertices depends on either the number of ertices or the number of edges. Therefore, there are seeral combinations. Roughly speaking, isoperimetric problems inoling edge-cuts correspond in a natural way to Cheeger constants in spectral geometry. The formulation and the proof techniques are ery similar. Cheeger constants were studied in the thesis of Cheeger [52], but they can be traced back to Polyá and Szegö [216]. We will follow tradition and call the discrete ersions by the same names, such as the Cheeger constant and the Cheeger inequalities. 23

2 24 2. ISOPERIMETRIC PROBLEMS 2.2. The Cheeger constant of a graph Before we discuss isoperimetric problems for graphs, let us first consider a measure on subsets of ertices. The typical measure assigns weight 1 to each erte, so the measure of a subset is its number of ertices. Howeer, this implies that all ertices hae the same measure. For some problems, this is appropriate only for regular graphs and does not work for general graphs. The measure we will use here takes into consideration the degree of a erte. For a subset S of the ertices of G, we define ol S, theolume of S, to be the sum of the degrees of the ertices in S: for S V (G). ol S = S d, Net, we define the edge boundary S of S to consist of all edges with eactly one endpoint in S, i.e., S = {{u, } E(G):u Sand S}. Let S denote the complement of S, i.e., S = V S. Clearly, S = S = E(S, S) where E(A, B) denotes the set of edges with one endpoint in A and one endpoint in B. Similarly, we can define the erte boundary δs of S to be the set of all ertices not in S but adjacent to some erte in S, i.e., δs = { S : {u, } E(G),u S}. We are ready to pose the following questions: Problem 1: Forafiednumberm, find a subset S with m ol S ol S such that the edge boundary S contains as few edges as possible. Problem 2: Forafiednumberm, find a subset S with m ol S ol S such that the erte boundary δs contains as few ertices as possible. Cheeger constants are meant to answer eactly the questions aboe. subset S V, we define E(S, (2.1) h G (S) = S) min(ol S,ol S). The Cheeger constant h G of a graph G is defined to be (2.2) h G =minh G (S). S For a In some sense, the problem of determining the Cheeger constant is equialent to soling Problem 1, since S h G ol S. We remark that G is connected if and only if h G > 0. We will only consider connected graphs. In a similar manner, we define the analogue of (2.1) for erte epansion (instead of edge epansion ). For a subset S V, we define (2.3) g G (S) = ol δ(s) min(ol S, ol S)

3 2.3. THE EDGE EXPANSION OF A GRAPH 25 and (2.4) g G =ming G (S). S For regular graphs, we hae g G (S) = δ(s) min( S, S ). We define for a graph G (not necessarily regular) and ḡ G (S) = δ(s) min( S, S ) ḡ G =minḡ G (S). S We remark that ḡ is the corresponding Cheeger constant when the measure for each erte is taken to be 1. More general measures will be considered later in Section 2.6. We note that both g G and ḡ G are concerned with the erte epansion of a graph and are useful for many problems The edge epansion of a graph In this section, we focus on the fundamental relations between eigenalues and the Cheeger constant. We first derie a simple upper bound for the eigenalue λ 1 in terms of the Cheeger constant of a connected graph. Lemma h G λ 1. Proof. We choose f based on an optimum edge cut C which achiees h G and separates the graph G into two parts, A and B: f() = 1 ol A if is in A, 1 if is in B. ol B By substituting f into (1.2), we hae the following: λ 1 C (1/ol A +1/ol B) 2 C min(ol A, ol B) = 2h G.

4 26 2. ISOPERIMETRIC PROBLEMS Now, we will proceed to gie a relatiely short proof of an inequality in the opposite direction, so that we will hae altogether 2h G λ 1 > h2 G 2. This is the so-called Cheeger inequality which often proides an effectie way for bounding the eigenalues of the graph. The following proof is one of four proofs of the Cheeger inequality and its ariations gien in [64]. Theorem 2.2. For a connected graph G, λ 1 > h2 G 2. Proof. We consider the harmonic eigenfunction f of L with eigenalue λ 1. We order ertices of G according to f. That is, relabel the ertices so that f( i ) f( i+1 ), for 1 i n 1. Let S i = { 1,..., i } and define α G =minh Si. i Let r denote the largest integer such that ol(s r ) ol(g)/2. Since g()d =0, g() 2 d =min (g() c) 2 d (g() g( r )) 2 d. c We define the positie and negatie part of g g( r ), denoted by g + and g, respectiely, as follows: { g() g(r ) if g() g( g + () = r ), 0 otherwise, { g() g(r ) if g() g( g ()= r ), 0 otherwise. We consider λ G = u (g(u) g())2 g()2 d u (g(u) g())2 (g() g( r)) 2 d ( (g+ (u) g + ()) 2 +(g (u) g ()) 2) ( g+ () 2 + g () 2). d u u Without loss of generality, we assume R(g + ) R(g ) and therefore we hae λ G R(g + ) since a + b c + d min{ a c, b d }. We here use the notation ol(s) =min{ol(s), ol(g) ol(s)} so that (S i ) α Gol(Si ).

5 2.3. THE EDGE EXPANSION OF A GRAPH 27 Then we hae λ G R(g + ) u = (g +(u) g + ()) 2 u g2 + (u)d u ( u = (g +(u) g + ()) 2)( u (g +(u)+g + ()) 2) u g2 + (u)d u u (g +(u)+g + ()) 2 ( u (g +(u) 2 g + () 2 ) ) 2 2 ( u g2 + (u)d ) 2 by the Cauchy-Schwarz inequality, u ( i = g +( i ) 2 g + ( i+1 ) 2 (S i ) ) 2 2 ( ) 2 by counting, u g2 +(u)d u ( i g +( i ) 2 g + ( i+1 ) 2 α G ol(s i ) ) 2 2 ( u g2 + (u)d ) 2 by the def. of α G, u ( = α2 G i g +( i ) 2 ( ol(s i ) ol(s i+1 ) ) ) 2 ( ) 2 2 u g2 +(u)d u ( i g ) +( i ) 2 2 d i = α2 G 2 = α2 G 2. ( u g2 + (u)d ) 2 u We hae proed that λ 1 h 2 G /2. There are seeral ways to show that the equality does not hold. One of the ways is to use the inequality in Theorem 2.3 (by noting > 2 /2for>0. This completes the proof of Theorem 2.2. We will state an improed ersion of Theorem 2.2 which howeer has a slightly more complicated proof. Theorem 2.3. For any connected graph G, we always hae λ G 1 1 h 2 G. Proof. From the proof of Theorem 2.2, we hae λ G (g + (u) g + ()) 2 u g+ 2 ()d = W.

6 28 2. ISOPERIMETRIC PROBLEMS Also, we hae W = ( (g + (u) g + ()) 2 ) ( + (u)+g + ()) u u (g 2 ) ( V g 2 + ()d ) ( u (g + (u)+g + ()) 2 ) ( g+(u) 2 g+() ) 2 2 u ( g 2 +()d ) (2 g 2 +()d W g 2 +()d ) ( i g 2 +( i ) g 2 +( i+1 ) (S i ) ) 2 (2 W )( g 2 +()) 2 d ( i (g+( 2 i ) g+( 2 i+1 ))α d j ) 2 j i (2 W )( g 2 + ())2 d This implies that Therefore we hae α 2 2 W. W 2 2W + α 2 0. λ 1 W 1 1 α h 2 G. For any connected (simple) graph G, wehae h G 2 ol G. Using Cheeger s inequality, we hae λ 1 > 1 2 ( 2 ol G )2 2 n 4. This lower bound is somewhat weaker than that in Lemma 1.9. Eample 2.4. For a path P n, the Cheeger constant is 1/ (n 1)/2. Asshown π in Eample 1.4, the eigenalue λ 1 of P n is 1 cos n 1 π2 2(n 1). This shows that 2 the Cheeger inequality in Theorem 2.2 is best possible up to within a constant factor. Eample 2.5. For an n-cube Q n, the Cheeger constant is 2/n which is equal to λ 1 (see Eample 1.6). Therefore the inequality in Lemma 2.1 is sharp to within a constant factor. Jerrum and Sinclair [169, 231] first used Cheeger s inequality as a main tool in deriing polynomial approimation algorithms for enumerating permanents and

7 2.4. THE VERTEX EXPANSION OF A GRAPH 29 for other counting problems. The reader is referred to [230] for the related computational aspects of the Cheeger inequality The erte epansion of a graph The proofs of upper and lower bounds for the modified Cheeger constant g G associated with erte epansion are more complicated than those for edge epansion. This is perhaps due to the fact that the definition of h G is in a way more natural and better scaled. Neertheless, erte epansion comes up often in many settings and it is certainly interesting in its own right. Since g G h G,wehae 2g G λ 1. For a general graph G, the eigenalue λ 1 can sometimes be much smaller than gg 2 /2. One such eample is gien by joining two complete subgraphs by a matching. Suppose n is the total number of ertices. The eigenalue λ 1 is no more than 8/n 2, but g G is large. Still, it is desirable to hae a lower bound for λ 1 in terms of g G.Herewegie a proof which is adapted from the argument first gien by Alon [5]. Theorem 2.6. For a connected graph G, gg 2 λ 1 >, 4d +2dg G where d denotes the maimum degree. Proof. We follow the definition in the proof of Theorem 2.2. We hae (f() f(u))f() λ 1 = > V + u V + d f 2 () (f() f(u)) u u, V + (g(u) g()) 2 u, g 2 ()d 2 + u, V + u V + V + d f 2 () f()(f() f(u)) Now we use the ma-flow min-cut theorem [129] as follows. Consider the network with erte set {s, t} X Y where s is the source, t is the sink, X is a copy of V + and Y is a copy of V (G). The directed edges and their capacities are gien as follows:

8 30 2. ISOPERIMETRIC PROBLEMS For eery in X, the directed edge (s, ) has capacity (1 + g G )d u where is labelled by erte u. For eery X, y Y, there is a directed edge (, y) with capacity d if is lablled by erte u, y is labelled by erte and {u, } is an edge. For eery X, y Y, there is a directed edge (, y) with capacity d u if and y are labelled by the same erte u. For eery y Y labelled by, the directed edge (y, t) has capacity d. To check that this network has its min-cut of size (1+g G )ol V +,letcdenote a cut separating s and t. Let X 1 = { X : {s, } C}and {Y = {y Y : {y, t} C}. Then C separates X 1 from Y \ Y. Therefore the total capacity of the cut C is at least the sum of capacities of the edges {s, },s X X 1,theedges(u, ), u X 1 and X 1 δx 1 \Y and edges (y, t), y Y. Since ol (X 1 δx 1 ) (1+g G )ol X 1, the total capacity of the cut is at least (1 + g G )ol(v + X 1 )+ol(x 1 δx 1 \ Y )+oly (1 + g G )ol(v + X 1 )+(1+g G )ol X 1 = (1+g G )ol V +. Since there is a cut of size (1 + g G )ol V +, we hae proed that the min-cut is of size equal to (1 + g G )ol V +. By the ma-flow min-cut theorem, there eists a flow function F (u, ) for all directed edges in the network so that F (u, ) is bounded aboe by the capacity of (u, ) andforeachfied Xand y Y,wehae F(, ) = (1+g G )d, Y F(, y) d y. X Then, 2 {u,} E = 2 {u,} E 2 F 2 (u, )(f + (u)+f + ()) 2 f 2 + ()( F 2 (u, )(f 2 +(u)+f 2 +()) u {u,} E F 2 (u, )+ f 2 +()(d 2 +(1+g G )dd ) u {u,} E F 2 (, u)) 2d(2 + g G ) f 2 + ()d.

9 2.5. A CHARACTERIZATION OF THE CHEEGER CONSTANT 31 Also, = u {u,} E f 2 + (u)( F (u, )(f 2 +(u) f 2 +()) {u,} E g G f+()d 2. F (u, ) {u,} E F (, u)) Combining the aboe facts, we hae (f + (u) f + ()) 2 λ 1 = {u,} E {u,} E ( f+()d 2 {u,} E (f + (u) f + ()) 2 F 2 (u, )(f + (u)+f + ()) 2 f+ 2 ()d {u,} E {u,} E F (u, )(f 2 +(u) f 2 +()) ) 2 F 2 (u, )(f + (u)+f + ()) 2 f+ 2 ()d 2d(2 + g G ) f+ 2 ()d ( F (u, )(f+ 2 1 (u) f + 2 ()) {u,} E 4d +2dg G f+ 2 ()d 2 g 2 G 4d +2dg G, as desired. Eample 2.7. For an n-cube, the erte isoperimetric problem has been well ( studied. According to the Kruskal-Katona theorem [174, 180], for a subset S of n ( k) ertices, for k n/2, the erte boundary of S has at least n k+1) ertices. Therefore, we hae g Qn = ( n n/2) 2 (n 1) 2 πn,forneen A characterization of the Cheeger constant In this section, we consider a characterization of the Cheeger constant which has similar form to the Rayleigh quotient but with a different norm.

10 32 2. ISOPERIMETRIC PROBLEMS (2.5) Theorem 2.8. The Cheeger constant h G of a graph G satisfies f() f(y) h G = inf sup f c R y f() c d V where f ranges oer all functions f : V R which are not constant functions. In language analogous to the continuous case, (2.5) can be thought of as f h G = inf sup. f c R f c and Proof. We choose c such that f()<c f() c If g = f c, thenforσ<0, we hae and for σ>0, we hae g()<σ g()<σ d d > d d d f()c d. f()>c d g()σ d. g()>σ We consider g(σ) = {{, y} E(G):g() σ<g(y)}. Then we hae f() f(y) = g(σ)dσ y = 0 h G dσ 0 g(σ) d g()<σ g()<σ dσ g()<σ = h G f() c d. V d + d dσ dσ g()>σ In the opposite direction, suppose X is a subset of V satisfying h G = E(X, X) ol X. g(σ) d d g()>σ g()>σ d

11 2.5. A CHARACTERIZATION OF THE CHEEGER CONSTANT 33 We consider a character function ψ defined by: { 1 if X, ψ() = 1 otherwise. Then we hae, ψ() ψ(y) sup C y V Therefore, we hae and Theorem 2.8 is proed. ψ() C d C = sup h G inf sup f c R 2 E(X, X) = 2ol X = h G. 2 E(X, X) (1 C)ol X +(1+C)ol X f() f(y) y f() c d V We will proe a ariation of Theorem 2.8 which is not sharp but seems to be easier to use. Later on it will be used to derie an isoperimetric relationship between graphs and their Cartesian products. Corollary 2.9. For a graph G, we hae f() f(y) h G inf f where f : V (G) R satisfies (2.6) y f() d V f()d =0. V Proof. From Theorem 2.8, we already hae f() f(y) h G inf f y f() d V 1 2 h G for f satisfying (2.6). It remains to proe the second part of the inequality. Suppose we define c as in the proof of Theorem 2.8. If c 0, then we hae f() c d f() d f() c = f() 0 f()0 f() d.

12 34 2. ISOPERIMETRIC PROBLEMS Therefore f() d 2 f() c d 2 f() c d. f() c The same results follows similarly if c 0. Thus we hae f() d 2 inf f() c d c and the desired upper bound on h G follows from f() f(y) inf f y 1 f() d 2 h G. V Suppose we decide to hae our measure be the number of ertices in S (and not the olume of S) forasubsetsof ertices. We can then pose similar isoperimetric problems. Problem 3: For a fied number m, what is the minimum edge-boundary for a subset S of m ertices? Problem 4: For a fied number m, what is the minimum erte-boundary for a subset S of m ertices? We can define a modified Cheeger constant, which is sometimes called the isoperimetric number, by h E(S, S) (S)= min( S, S ) and h G = inf S h (S). We note that h G min d h G h G ma d. These modified Cheeger constants are related to the eigenalues of L, denoted by 0 = λ 0 λ 1 λ n 1,and (f(u) f()) 2 λ 1 = inf sup f c = inf f u f,lf f,f (f() c) 2 where f ranges oer all functions f satisfying f() = 0 which are not identically zero. The aboe definition differs from that of L in (1.3) by the multiplicatie factors of d for each term in the sum of the denominator. So, eigenalues λ i of L satisfy 0 λ i λ i ma d.

13 2.6. ISOPERIMETRIC INEQUALITIES FOR CARTESIAN PRODUCTS 35 By using methods similar to those in preious sections, we can show 2h G λ 1. Howeer, the lower bound for λ 1 in terms of h G is a little messy in its deriation. We need to use the fact: u (f(u)+f()) 2 2 f() 2 d 2 f() 2 ma d w w in order to derie the modified Cheeger inequality: λ 1 h 2 G 2ma d. This is less elegant than the statement in Theorem 2.2. We remark that the erte epansion ersion of the Cheeger inequality are closely related to the so-called epander graphs, which we will eamine further in Chapter Isoperimetric inequalities for Cartesian products Suppose G is a graph with a weight function w which assigns nonnegatie alues to each erte and each edge. A general Cheeger constant can be defined as follows: w(, y) h(g, w) =min S {,y} E(S, S) min( S w(), y S w(y)). We say the weight function w is consistent if w(u, ) =w(). u For eample, the ordinary Cheeger constant is obtained by using the weight function w 0 () =d for any erte and w 0 (u, ) = 1 for any edge {u, }. Clearly, w 0 is consistent. On the other hand, the modified Cheeger constant is h G = h(g, w 1) where the weight function w 1 satisfies w 1 (u, ) = 1 for any edge {u, } and w 1 () = 1 for any erte. In this case, w 1 is not necessarily consistent. We note that graphs with consistent weight functions correspond in a natural way to random walks and reersible Marko chains. Namely, for a graph with a consistent weight function w, we can define the random walk with transition probability of moing from a erte u to each of its neighbors to be P (u, ) = w(u, ) w(). Similar to Theorem 2.8, the general isoperimetric inariant h(g, w) has the following characterization:

14 36 2. ISOPERIMETRIC PROBLEMS Theorem For a graph G with weight function w, the isoperimetric inariant h(g, w) of a graph G satisfies f() f(y) w(, y) (2.7) h(g, w) = inf sup f c R y f() c w() where f ranges oer all f : V R which are not constant functions. V In particular, we also hae the following characterization for the modified Cheeger constant. Theorem f() f(y) (2.8) h G = inf sup f c R y f() c where f ranges oer all f : V R which are not constant functions. V For two graphs G and H, thecartesian product G2H has erte set V (G) V (H) with(u, ) adjacent to (u, ) if and only if u = u and is adjacent to in H, or= and u is adjacent to u in G. For eample, the Cartesian product of n copies of one single edge is an n-cube, which is sometimes called a hypercube. The isoperimetric problem for n-cubes is an old and well-known problem. Just as in the continuous case where the sets with minimum erte boundary form spheres, in a hypercube the subsets of gien size with minimum erte-boundary are socalled Hamming balls, which consist of all ertices within a certain distance [23, 158, 159, 191]. The isoperimetric problems for grids (which are Cartesian products of paths) and tori (which are Cartesian products of cycles) hae been well-studied in many papers [34, 35, 252]. We also consider a Cartesian product of weighted graphs with consistent weight functions. For two weighted graphs G and G, with weight functions w, w, respectiely, the weighted Cartesian product G G hasertesetv(g) V(G )with weight function w w defined as follows: For an edge {u, } in E(G), we define w w ((u, ), (, )) = w(u, )w ( ) and for an edge {u, } in E(G ), we define w w ((u, u ), (u, )) = w(u)w (u, ). We require w w to be consistent. Clearly, foraerte=(u, ) ing G, the weight of in G G is eactly 2w(u)w (). In general, for graphs G i with consistent weight functions w i, i =1,...,k,the weighted Cartesian product G 1 G k has erte set V (G) V(G k )with a consistent weight function w 1 w k defined as follows: For an edge {u, } in E(G i ), the edge joining ( 1,..., i 1,u, i+1,..., k )and( 1,..., i 1,, i+1,..., k )hasweightw 1 ( 1 )...w i 1 ( i 1 )w i (u, )w i+1 ( i+1 )...w k ( k ). We remark that G 1 G 2 G 3 is different from (G 1 G 2 ) G 3 or G 1 (G 2 G 3 ). The weighted Cartesian product of graphs corresponds naturally to the Cartesian product of random walks on graphs. Suppose G 1,...,G k are graphs with the erte sets V (G i ). Each G i is associated with a random walk with transition

15 2.6. ISOPERIMETRIC INEQUALITIES FOR CARTESIAN PRODUCTS 37 probability P i as defined as in Section 1.5. The Cartesian product of the random walks can be defined as follows: At the erte ( 1,..., k ), first choose a random direction i, between 1 and k, each with probability 1/k. Then moe to the erte ( 1,..., i 1,u i, i+1,..., k ) according to P i. In other words, P (( 1,..., i 1, i, i+1,...,u k ),( 1,..., i 1,u i, i+1,..., k )) = 1 k P ( i,u i ). We point out that the aboe two notions of the Cartesian products are closely related. In particular, cλ G2H λ G H c 1 λ G2H where min (min deg G, min deg H) c = ma (ma deg G, ma deg H). Here min deg and ma deg denote the minimum degree and the maimum degree, respectiely. The random walk on G 1 2 2G k has transition probability P of moing from a erte ( 1,..., k )totheerte( 1,..., i 1,u i, i+1,..., k )gien by: P (( 1,..., i 1, i, i+1,...,u k ),( 1,..., i 1,u i, i+1,..., k )) = w( i,u i ) w( j ). 1 j k For a graph G, the natural consistent weight function associated with G has edge weight 1 and erte weight d for any erte. Then we hae the following. Theorem The eigenalue of a weighted Cartesian product of G 1,G 2,..., G k satisfies λ G1 G 2 G k = 1 k min(λ G 1,λ G2,...,λ Gk ) where λ G denotes the first eigenalue λ 1 of the graph G. Here we will gie a proof for the case k = 2. Namely, we will show that the eigenalue of a weighted Cartesian product of G and H satisfies (2.9) λ G H = 1 2 min(λ G,λ H ). Proof. Without loss of generality, we assume that λ G λ H. It is easy to see that λ G H 1 2 λ G. Suppose f : V (G) R is the harmonic eigenfunction achieing λ G. We choose a function f 0 : V (G) V (H) R by setting f 0 (u, ) =f(u). Clearly, λ G H is less than the Rayleigh quotient using f 0 whose alue is eactly λ G /2.

16 38 2. ISOPERIMETRIC PROBLEMS In the opposite direction, we consider the harmonic eigenfunction g : V (G) V (H) R achieing λ G H. We denote, for u V (G), V(H), g(u, )d g u = g =, ol H g(u, )d u u, ol G g(u, )d u d u, (2.10) c = ol G ol H. Here, we repeatedly use the definition of eigenalues and the Cauchy-Schwarz inequality: (g(u, ) g(u,)) 2 d + (g(u, ) g(u, )) 2 d u u u λ G H = u (g(u, ) c) 2 2d u d λ G u, (g(u, ) g ) 2 d u d +( u, u (g(u, ) g ) 2 2d u d + u, u, λ G λ G 2. This completes the proof of (2.9). d u ) (g g ) 2 (g c) 2 2d u d d u ) (g c) 2 d (g(u, ) g ) 2 d u d + λ H ( u, u (g(u, ) g ) 2 2d u d + (g c) 2 2d u d u, u, Theorem The Cheeger constant of a weighted Cartesian product of G 1,G 2,,G k satisfies 1 k min(h G 1,h G2,...,h Gk ) h G1 G 2 G k 1 2k min(h( G 1,h G2,...,h Gk ). Here we again will proe the case for the product of two graphs and leae the proof of the general case as an eercise. 1 (2.11) 2 min(h G,h H )h G H 1 4 min(h G,h H ). Proof. Without loss of generality, we assume that h G h H.

17 2.6. ISOPERIMETRIC INEQUALITIES FOR CARTESIAN PRODUCTS 39 First we note that h G H h G 2. Suppose f : V (G) R is a function achieing h(g) in (2.7). We choose a function f 0 : V (G) V (H) R by setting f 0 (u, ) =f(u). Clearly, h G H is no more than the alue for the quotient of (2.7) using f 0 whose alue is eactly h G /2. It remains to show that h G H h G /4. To this end, we will repeatedly use Corollary 2.9, and we adopt the notation in the proof of (2.9). h G H = g(u, ) g(u,) d + g(u, ) g(u, ) d u u u u g(u, ) c 2d u d u, h G g(u, ) g d u d +( d u ) g g u, u g(u, ) g 2d u d + g c 2d u d u, u, h G g(u, ) g d u d + h H 2 2 ( d u ) g c d u, u g(u, ) g 2d u d + g c 2d u d u, u, h G 4. This completes the proof of (2.11). For the modified Cheeger constant h G, a similar isoperimetric inequality can be obtained: Corollary The modified Cheeger constant of the Cartesian product of G 1,G 2,...,G k satisfies min(h G 1,h G 2,...,h G k ) h G 12G 22 2G k 1 2 min(h G 1,h G 2,...,h G k ). Notes The proof is quite similar to that of (2.11) (also see [87]) and will be omitted. The characterization of the Cheeger constant in Theorem 2.8 is basically the Rayleigh quotient using the L 1 -norm both in the numerator and denominator. In

18 40 2. ISOPERIMETRIC PROBLEMS general, we can consider the so-called Sobole constants for all p, q > 0: ( ) 1/p f(u) f() p s p,q = inf f = inf f u ( ) 1/q f() q d f p f q where f ranges oer functions satisfying f() c q d f() q d for any c, or, equialently, f c q f q. The eigenalue λ 1 is associated with the case of p = q = 2, while the Cheeger constant corresponds to the case of p = q = 1. Some of the general cases will be considered later in Chapter 11 on Sobole inequalities. This chapter is mainly based on [63]. More general cases of the Cartesian products are discussed in [87]. Another reference for weighted Cheeger constants and related isoperimetric inequalities is [83].