Lower bounds for the isoperimetric numbers of random regular graphs
aa r X i v : . [ m a t h . C O ] F e b LOWER BOUNDS FOR THE ISOPERIMETRIC NUMBERS OF RANDOMREGULAR GRAPHS
BRETT KOLESNIK AND NICK WORMALDA
BSTRACT . The vertex isoperimetric number of a graph G = ( V , E ) is the min-imum of the ratio | ∂ V U | / | U | where U ranges over all nonempty subsets of V with | U | / | V | ≤ u and ∂ V U is the set of all vertices adjacent to U but not in U . The analogously defined edge isoperimetric number—with ∂ V U replacedby ∂ E U , the set of all edges with exactly one endpoint in U —has been stud-ied extensively. Here we study random regular graphs. For the case u = d ≥
3. Moreover, we obtain a lower bound on the asymp-totics as d → ∞ . We also provide asymptotically almost sure lower bounds on | ∂ E U | / | U | in terms of an upper bound on the size of U and analyse the boundsas d → ∞ .
1. I
NTRODUCTION
In this paper we consider versions of the isoperimetric number of randomregular graphs. These are explicit indicators of the notion generally called ex-pansion (see below for the relevant definitions). Random regular graphs givenondeterministic examples of expander graphs, and as mentioned in [12, Sec-tion 4.6], there is great interest in the edge and vertex expansion of sets of vary-ing sizes. Here we obtain explicit bounds on the expansion of sets with givensize in random regular graphs. We concentrate on the vertex version, which ismore difficult and less well studied than the edge version.Let G be a graph on n vertices. For a subset U of its vertex set V = V ( G ),let ∂ V ( U ) denote the set of all vertices adjacent to a vertex in U but not in U .Similarly, let ∂ E ( U ) denote the set of all edges with exactly one end in U . Notethat | ∂ V ( U ) | ≤ | ∂ E ( U ) | . For any 0 < u ≤ u-edge isoperimetric number isdefined as i E , u ( G ) = min | U |≤ un | ∂ E ( U ) | / | U | ,and likewise for any 0 < u ≤ u-vertex isoperimetric number of G as i V , u ( G ) = min | U |≤ un | ∂ V ( U ) | / | U | .The 1/2-edge (1/2-vertex) isoperimetric number is often referred to as the edge (vertex) isoperimetric number , and in this case, we simplify notation as i E ( G ) and i V ( G ). For the edge version, this makes immediate sense since a lowerbound on the number of edges joining S ⊆ V to V \ S is obtained as i E ( G ) ρ where ρ = min{ | S | , | V \ S | }. For the vertex isoperimetric number, the situation is notquite so symmetrical since | ∂ V ( U ) | 6= | ∂ V ( V \ U ) | in general. However, u = B. KOLESNIK AND N. WORMALD has some uses. Note that the iterated neighbourhoods of any vertex in G expandby (at least) a factor α = + i V ( G ) until they reach size n /2 (or more). Hence aneasy upper bound on the diameter of G is 2 log α ( n /2). To give another example,Sauerwald and Stauffer [17] recently showed that if a certain rumour spreadingprocess takes place on a regular graph, where informed vertices randomly se-lect a neighbour to inform (i.e., the push model), then all vertices are informedasymptotically almost surely (a.a.s.) after O ((1/ i V ) · log n ) steps of the process.Hence a lower bound on i V gives a upper bound (holding a.a.s.) for the time atwhich all vertices are aware of the rumour.A graph is d -regular if all its vertices are of degree d . Let G n , d denote theuniform probability space on the set of all d -regular graphs on n vertices thatare simple (i.e., have no loops or multiple edges). A property holds a.a.s. in asequence of probability spaces on { Ω n } if the probability that an element of Ω n has the given property converges to 1 as n → ∞ . Define i V , u ( d ) = sup © ℓ : i V , u ( G n , d ) ≥ ℓ a.a.s. ª and define i E , u ( d ) similarly. In the case u = i V ( d ) and i E ( d ).We can now describe our results more explicitly. Our main purpose is to pro-vide asymptotically almost sure lower bounds for the vertex expansion of ran-dom regular graphs.In Section 2, we highlight some results in the literature that relate to vertexand edge expansion of regular graphs.The pairing model, as used by Bollobás [5] to investigate i E ( d ), i.e., for thecase of edge expansion, is discussed in Section 3. As we shall see, this model isalso helpful for studying i V , u ( d ).In Section 4, we introduce the method we use. In short, we obtain lowerbounds on vertex expansion using the first moment method: for a sequenceof non-negative, integer-valued random variables { X n }, provided that E ( X n ) → n → ∞ , it follows that X n = i E ( d ). However, in bounding i V , u ( d ), the method yields bounds which are initially quite opaque. The maincomplication is that both ∂ V and ∂ E need to be considered. For a sequence u ( n ) → u ∈ [0, 1/2] and numbers s and y , we define a random variable X thatcounts the number of subsets U with | U | = un , | ∂ V | = sn and | ∂ E | = yn in a graphfrom G n , d . For such a sequence, we use the first moment method to determinethe range of s for which X = u-vertex expansionnumber I V , u ( d ), which can be thought of as an asymptotic profile for the vertexexpansion of subsets U ⊂ V ( G n , d ) with | U | ∼ un . (See Section 4 for the precisedefinition.)We obtain lower bounds A d ( u ) on I V , u ( d ) in Section 4, and in Table 1 we pro-vide approximate values for A d ( u ) for several values of d and u . Since one ex-pects the isoperimetric number of a graph to be obtained by larger sets, it is rea-sonable to conjecture that the u -isoperimetric and expansion numbers coincide,i.e., i V , u ( d ) = I V , u ( d ), and hence, for all d ≥ u ∈ (0, 1/2], i V , u ( d ) ≥ A d ( u ). SOPERIMETRIC NUMBERS OF RANDOM REGULAR GRAPHS 3
Unfortunately, the formulae do not seem to have a convenient explicit form, andso, this is not straightforward to show for the cases u < i V , u ( d ) for u < u = i V ( d ) for all d ≥ Theorem 1.
For d ≥ , i V ( d ) ≥ A d (1/2) = s d where s d is the smallest positive solution to (2 d − s = d /2 + s − (1 − s ) − s s s .Table 2 provides approximate values for A d (1/2) for several values of d .In Section 6 we apply Theorem 1 to obtain a lower bound on the asymptoticsof i V ( d ) as d → ∞ . Corollary 2.
As d → ∞ , i V ( d ) ≥ − d + O ((log d )/ d ).Corollary 2 improves upon the information that is otherwise available fromspectral results. See Section 2 for a discussion on this.We switch our attention to the edge isoperimetric number in Section 7. Bol-lobás [5] computed lower bounds with the first moment method for i E ( d ), i.e.,for the case of edge expansion at u = d ≥
3. Therein it is shown thatfor sufficiently large d , i E ( d ) ≥ d /2 − q d log 2 (1)and so lim d →∞ i E ( d )/ d = n ≥ d +
1, if G is a d -regular graph in n vertices and U is selected uniformly from {1, 2, . . . , n } such that | U | = ⌊ n /2 ⌋ , then E ( ∂ V U ) = d ⌊ n /2 ⌋⌈ n /2 ⌉ /( n − d . Then, more generally, it is claimed in [5] thatlim d →∞ i E , u ( d )/ d = − u for all 0 < u < i E , u ( d ) for all d ≥ < u ≤ u-edge expansion numberI E , u ( d ) (analogous to I V , u ( d )) yields a simple method for computing ‘best possi-ble’ lower bounds on i E , u ( d ) for all d ≥ < u ≤ B. KOLESNIK AND N. WORMALD
Theorem 3.
For d ≥ and u ∈ (0, 1/2] ,i E , u ( d ) ≥ b A d ( u ) = y d , u / uwhere y d , u is the smallest positive solution tod d /2 u ( d − u (1 − u ) ( d − − u ) = ( d u − y ) ( du − y )/2 ( d − d u − y ) ( d − du − y )/2 y y .Table 3 provides approximate values for b A d ( u ) for several values of d and u .Applying Theorem 3, in Section 7 we obtain lower bounds on the asymptoticsof i V , u ( d ), as d → ∞ , for all 0 < u ≤ Corollary 4.
Fix u ∈ (0, 1/2] . As d → ∞ , b A d ( u ) = d (1 − u ) − − u ) q d log( u − u (1 − u ) u − ) + o ( p d ). Hence i E , u ( d ) ≥ d (1 − u + o (1)) ( as d → ∞ ).2. B ACKGROUND A u -edge ( u -vertex) α -expander, α >
0, is a graph that has a u -edge ( u -vertex)isoperimetric number at least as large as α . The u -isoperimetric numbers of agraph may be viewed as indicators for its level of connectivity. Expander graphsprovide a wealth of theoretical interest and have many applications. For a thor-ough exposition of the theory and applications of expander graphs, see [12] andthe references therein.Regular graphs are known to be good expanders with high probability; how-ever, determining the isoperimetric number with precision is a difficult task. Forinstance, as shown by Golovach [11], even the problem of determining whether i E ( G ) ≤ q , provided q ∈ Q and G has degree sequence bounded by 3, is NP -complete. Thus, bounds for the isoperimetric numbers of regular graphs areof interest.Buser [6] showed that for all n ≥ n vertices whose edge isoperimetric number is at least 1/128. To quote Bol-lobás [5]:Buser’s proof . . . was very unorthodox in combinatorics and veryexciting: it used the spectral geometry of the Laplace operator onRiemann surfaces, Kloosterman sums and the Jacquet–Langlandstheory. As Buser wrote: the proof ‘is rather complicated and itwould be more satisfactory to have an elementary proof.’Using the standard first moment method, Bollobás [5] provided a simple proofthat much more is true. In [5] it is shown that in fact i E ( d ) ≥ d /2 − q d log 2 (as d → ∞ ).Several bounds for small d are provided in [5], such as i E (3) > i E (4) > i E (5) > λ ( G ) denote the second largest SOPERIMETRIC NUMBERS OF RANDOM REGULAR GRAPHS 5 (i.e., largest other than d ) eigenvalue in the adjacency matrix of G . Alon andMillman [2] proved that if G is d -regular, then i E ( G ) ≥ ( d − λ )/2.Further, Alon [1] showed if n > d and G is d -regular, then i E ( G ) ≤ d /2 − p d /16 p i E (3) ≥ d > i E ( d ) givenin [5] by investigating subsets which are, in a certain sense, locally optimal. Up-per bounds for i E ( d ) are available via results for the bisection width of regulargraphs. Note that the (edge) bisection width of a graph G = ( V , E ) is defined as b ( G ) = min ( n − ≤| U |≤ n /2 | ∂ E ( U ) | / | U | ,and so clearly for any graph G , i E ( G ) ≤ b ( G ).The best upper bound for cubic graphs is that of Monien and Preis [15]. Thereinit is proved b ( G ) ≤ + ǫ for all ǫ > all sufficiently large connected cubicgraphs G . (The required lower bound on the size of G depends on ǫ .) Hence i E (3) ≤ d > i E (4) ≤ i E (5) ≤ i E (6) ≤ i E (7) ≤ G is d -regular, then i V , u ( G ) ≤ i E , u ( G ) ≤ d · i V , u ( G )for all 0 < u ≤ u .Some interesting results are as follows. Tanner [18] proved that if G ∈ G n , d and λ ( G ) ≤ α d , then i V , u ( G ) ≥ ¡ u (1 − α ) + α ¢ − λ ( G ) ≤ p d − + ǫ a.a.s. in G n , d for any ǫ >
0. Thus it follows for any d ≥ < u ≤ i V , u ( d ) ≥ ¡ u (1 − d − d ) + d − d ¢ − i V ( d ) ≥ − d + O (1/ d ). (2) B. KOLESNIK AND N. WORMALD
Finally, one other result of interest concerns the expansion of small sets; see [12,Theorem 4.16.1]. For any d ≥ δ >
0, there exists an ǫ δ > i E , ǫ δ ( G ) ≥ d − − δ (3)a.a.s. in G n , d . In fact, the same is true of vertex expansion. In what follows, werefer to the above as the small sets property . In both cases, the expansion param-eter d − k ,min | U |= k | ∂ V U | / | U | ≤ min | U |= k | ∂ E U | / | U | ≤ d − + k . (4)For details see [12, Subsection 5.1.1].3. M ODEL FOR ANALYSIS
To analyse G n , d we make use of the pairing model, P n , d , described as follows.Suppose there are n cells, each containing d points, where d n is even. Let P n , d denote the uniform probability space on the set of all perfect matchings of the d n points. By collapsing each cell of a given H ∈ P n , d into a single vertex, a d -regular multigraph π H on n vertices is obtained. The pairing model is due toBollobás, who was the first to suggest directly deducing properties of randomgraphs from the model, though similar models appear in earlier works (see [19]for details).It is known that P ( π H is simple) is bounded away from 0 as n tends to infinity,and that all d -regular simple graphs are selected with equal probability throughthe process of choosing an H ∈ P n , d uniformly at random and then constructing π H . Thus, to prove that a property occurs a.a.s. in G n , d , it is enough to provethat the pairings in P n , d a.a.s. have the corresponding property. A survey ofproperties of random d -regular graphs proved using this model is in [19].We make isoperimetric definitions for pairings to coincide with the same pa-rameters for the corresponding (multi)graphs. For a pairing H ∈ P n , d and a sub-set U of its n cells, let ∂ V ( H ) ( U ) denote the set of all cells adjacent to a cell in U but not in U . Similarly, let ∂ E ( H ) ( U ) denote the set of all edges with exactly oneendpoint in a cell of U . Note that | ∂ V ( H ) ( U ) | ≤ | ∂ E ( H ) ( U ) | . For any 0 < u ≤
1, the u -vertex isoperimetric number for a pairing H ∈ P n , d is defined as i V , u ( H ) = min | U |≤ un | ∂ V ( H ) ( U ) | / | U | ,and likewise for any 0 < u ≤ u -edge isoperimetric number is defined as i E , u ( H ) = min | U |≤ un | ∂ E ( H ) ( U ) | / | U | .Furthermore, put i V , u ( P , d ) = sup © ℓ : i V , u ( P n , d ) ≥ ℓ a.a.s. ª and define i E , u ( P , d ) analogously. SOPERIMETRIC NUMBERS OF RANDOM REGULAR GRAPHS 7
4. L
OWER BOUNDS FOR VERTEX EXPANSION
For a sequence u = u ( n ) with 0 < u ≤ n , we define the u-vertex expan-sion number to be I V , u ( d ) = sup ½ ℓ : min U ⊂ V , | U |= un | ∂ V U | un ≥ ℓ a.a.s. in G n , d ¾ .The motivation to study the vertex expansion number is the following relationto the isoperimetric number. Lemma 5.
Fix < u ≤ . Theni V , u ( d ) ≥ inf ≤ u ≤ u inf w → u I V , w ( d ), where the second infimum is over sequences w ( n ) with < w ≤ .Proof. Set L to be right-hand side of the inequality, and assume by way of con-tradiction that, for some ǫ > i V , u ( d ) is not at least L − ǫ a.a.s. Then for all n insome infinite set S of positive integers, and some ǫ ′ >
0, we have P ( i V , u ( G n , d ) < L − ǫ ) > ǫ ′ . Thus, there exists a function w ( n ) > u ∈ [0, u ]such that in G n , d P µ min | U |= w ( n ) n | ∂ V ( U ) || U | < L − ǫ ¶ > ǫ ′ for all n ∈ S . For any sequence w ′ ( n ) → u with w ′ ( n ) = w ( n ) for all n ∈ S , we havethat I V , w ′ ( d ) ≤ L − ǫ , giving the desired contradiction. (cid:4) Of course we expect I V , u ( d ) ≥ i V , u ( d ) if u → u > f suchthat I V , u ≥ f ( c ) when u ( n ) ∼ c . From our argument, we will be able to concludethat for fixed u , i V , u ≥ min{ f ( c ) : 0 ≤ c ≤ u }.To analyse I V , u ( d ) it will be useful to first look at the analogously defined quan-tity I V , u ( P , d ) for pairings.The main complication in bounding I V , u ( P , d ) via the first moment methodis that both ∂ V and ∂ E must be considered to compute the expected number ofelements of P n , d with i V , u equal to some specified value. Consequently, bound-ing I V , u is more involved than I E , u , as in the latter case we need only take ∂ E intoaccount.For a randomly selected element of P n , d , let X ( n ) u , s , y , d denote the number ofsubsets of V of size un that have | ∂ V | = sn and | ∂ E | = yn . Here s , u and y will befunctions of n . Let C n , s , y denote the coefficient of x yn in the polynomial à d X j = à dj ! x j ! sn = ¡ ( x + d − ¢ sn , (5)so that C n , s , y is the number of ways to choose yn elements of sd n items parti-tioned into sn groups of cardinality d each, such that at least one item is chosen B. KOLESNIK AND N. WORMALD from each group. Note also that M (2 m ) = (2 m )!/ m !2 m is the number of perfectmatchings of 2 m points, so, for instance, | P n , d | = M ( d n ). Then in P n , d , E ¡ X ( n ) u , s , y , d ¢ = C n , s , y à nun !à n − unsn !à d unyn ! ( yn )! M ( d un − yn ) M ( d n − d un − yn ) M ( d n ) ,where the binomial coefficients choose a set U of un vertices, their sn neigh-bours, and the yn points inside them that join to points outside U , and the otherfactors count choices of the pairs with the obvious restrictions. For any x > C n , s , y ≤ x − yn ¡ ( x + d − ¢ sn .We will use this upper bound for various x > C n , s , y .By Stirling’s approximation, for any x > ¡ E X ( n ) u , s , y , d ¢ n ≤ ( d u ) du ( d − d u − y ) ( d − du − y )/2 ¡ ( x + d − ¢ s φ ( n ) x y u u s s (1 − u − s ) (1 − u − s ) ( d u − y ) ( du − y )/2 d d /2 ,where φ ( n ) = n O (1/ n ) contains factors which are of polynomial size before takingthe n th root. Hence for x >
0, we havelog E ¡ X ( n ) u , s , y , d ¢ ≤ n ¡ f d ( u , s , y , x ) + o (1) ¢ (6)where f d ( u , s , y , x ) = d u log( d u ) + ( d − d u − y )(log( d − d u − y ))/2 + s log ¡ ( x + d − ¢ − y log x − u log u − s log s − (1 − u − s ) log(1 − u − s ) − ( d u − y )(log( d u − y ))/2 − ( d log d )/2.One particular value we will use is x = x , defined as the value at which ∂ f / ∂ x =
0, or equivalently sd ( x + d − ( x + d − = yx . (7)This choice of x is important, since if ( y − s ) n → ∞ and ( sd − y ) n → ∞ as n → ∞ and x > x − y ¡ ( x + d − ¢ s ∼ C nn , s , y (as n → ∞ ). (8)Since the relevant range of y is s ≤ y ≤ d s (as for any U ⊂ V , | ∂ V U | ≤ | ∂ E U | ≤ d · | ∂ V U | ), it can be observed by (8) that using x = x to bound C n , s , y in our argu-ment leads to just as good final results as using the precise formula.We briefly outline two arguments for the asymptotics at (8). One simple op-tion is to use the limit theorems of Bender [4, Theorems 3 and 4] (cf. [16] and [7]).Another more transparent way to obtain (8) is as follows: let Y i be i.i.d. ran-dom variables with support {0, 1, . . . , d −
1} and taking values i with probability ¡ di + ¢ /(2 d − Y ( sn ) = P sni = Y i , and observe C n , s , y = P ( Y ( sn ) = yn ). SOPERIMETRIC NUMBERS OF RANDOM REGULAR GRAPHS 9
Log-concave sequences are unimodal. Let y ∗ n denote an exponent associatedwith the coefficient in the sn th convolution of p ( x ) attaining (as close as possi-ble to) the centre of mass. By the Berry-Esseen inequality, Y ( sn ) is asymptot-ically normal. Hence the asymptotics of C n , s , y ∗ can be established. Moreover,the asymptotics of an arbitrary coefficient, C n , s , y , say, may be obtained as fol-lows. For any r ∈ R , we have r m [ x m ] p ( x ) n = [ x m ] p ( xr ) n ,where [ x m ] g ( x ) denotes the coefficient of x m in the polynomial g . Observe thatsince p ( x ) has log-concave coefficients, so does p ( xr ), and it then follows by awell-known property that the same holds for p ( xr ) n . Hence, selecting r suchthat the centre of mass in p ( xr ) n is located at the coefficient of x m , the asymp-totics of [ x m ] p ( x ) n can be found by [4, Lemma 1].Note that f is continuous on the natural domain in question, with the con-vention 0 log 0 =
0. Define M d ( u , s , y ) = min x ≥ f d ( u , s , y , x ), h d ( u , s ) = max s ≤ y ≤ min{ ds , du } M d ( u , s , y )and H d ( u ) = min © s : h d ( u , s ) ≥ ª .A little examination of f shows that the various min’s and max’s exist and arecontinuous. Recalling that every relevant x leads to an upper bound in (6), wemay now deduce the following. Lemma 6. H d has the following properties. (a) H d ( u ) = if and only if u ∈ {0, 1} . (b) Fix < u < . If u = u ( n ) → u as n → ∞ , thenI V , u ( d ) ≥ H d ( u ) u . In the case that u → + , I V , u ( d ) ≥ d − For any < u < , we havei V , u ( d ) ≥ inf < u ≤ u H d ( u ) u . Proof.
For part (a), note that H d ( u ) = u ∈ {0, 1}. Observe that f d (0, 0, 0, · ) = = f d (1, 0, 0, · ) (noting that when s =
0, the only possible value y inthe max function in the definition of h d ( u , s ) is 0, and then x does not appear in f ). So h d (0, 0) = h d (1, 0) ≥
0, and hence H d (0) = H d (1) =
0. Conversely, suppose H d ( u ) =
0. Then h d ( u , 0) ≥
0, and so we have f d ( u , 0, 0, · ) ≥
0. Thus, sinced f d ( u , 0, 0, · )d u = d − u (1 − u ) > < u < u is either 0 or 1. For part (b), consider 0 < u < u ( n ) → u . We have H d ( u ) > s be positive with s < H d ( u ). Then h d ( u , s ) <
0. By definition, h d ( u , s ) is nondecreasing in s . So, by continuity, for n sufficiently large, h d ( u , s ) < h d ( u , s )/2 for all s ≤ s . Then (6) tells us that in P n , d , the expected number ofsets U with | U | = un , | ∂ V U | = sn ≤ s n , and | ∂ E U | = yn is exponentially smallfor every relevant y (noting that y ≥ s can be assumed because every boundaryvertex has at least one boundary edge). Summing over all O ( n ) relevant valuesof s and y , we deduce by the union bound that a.a.s. | ∂ V U | ≥ s n for all U with | U | = un . Hence, I V , u ( P , d ) ≥ H d ( u )/ u .Thus, the first inequality in (b) follows in view of the relation between G n , d and P n , d discussed in Section 3.In the case that u → + , we use the small sets property discussed at (3): for all δ >
0, there is some N δ ∈ N so that for n > N δ , u ( n ) < ǫ δ ( ǫ δ as guaranteed by theproperty), and hencemin | U |= u ( n ) n | ∂ V U || U | ≥ min | U |≤ ǫ δ n | ∂ V U || U | ≥ d − − δ a.a.s. in G n , d . The above holds for any δ >
0, so we have I V , u ( d ) ≥ d − i V , u ( d ) ≥ min ½ inf w → I V , w ( d ), inf < u ≤ u H d ( u )/ n ¾ and the first of these is at least d − i V , u ( d ) ≤ d − u >
0, and (c) follows. (cid:4)
Of course, this result is best possible for a direct application of the first mo-ment method, in the sense that, from (6) and the earlier discussion, the expectednumber of sets of size s with a boundary which is slightly larger than H d ( u ) can-not be exponentially small.Now we need to discuss the behaviour of f d ( u , s , y , x ), which we often abbre-viate to f . Similarly, since d and u are fixed for the whole discussion, we oftenrefrain from explicitly mentioning them as parameters of other functions.A ‘direct attack,’ solving for the minimum x and maximum y in the definitionof h d ( u , s ), and then analysing its behaviour as a function of s , leads to calcula-tions that seem too complicated. Instead we take an indirect approach: for eachsuitable y , we will compute a value of s such that y is the maximiser. Even withthis, we do not restrict ourselves to using the minimising x , which, considering ∂ f / ∂ x , turns out to be x . For an upper bound, we are free to choose any x . Tosimplify the argument we will, for part of it, use a different choice of x , whichhappens to coincide with x everywhere that matters.The relevant partial derivatives are ∂ f ∂ x = sd ( x + d − ( x + d − − yx SOPERIMETRIC NUMBERS OF RANDOM REGULAR GRAPHS 11 and ∂ f ∂ y = log ˆ x ( y ) − log x ,where ˆ x ( y ) = s d u − yd − d u − y for all y < d u .For any x and y , define S ( y , x ) = y ¡ ( x + d − ¢ xd ( x + d − (9)so that s = S ( y , x ) satisfies (7), and setˆ s ( y ) = S ( y , ˆ x ( y )) (10)and F ( y ) = F d , u ( y ) = f d ( u , ˆ s ( y ), y , ˆ x ( y )). (11) Lemma 7.
Fix < u ≤ . We have d ˆ s ( y )/d y > .Proof. This follows easily by checking that ˆ x ( y ) is a nonincreasing function of y and that ∂ S ( x , y )/ ∂ x < + d x − ( x + d < (cid:4) We are now ready to state our main result for this section. From now on, werestrict attention to u ≤ u . We say that a real-valued function g with a real domain D is unimodal with mode ˜ y if g ( y ) is strictly increasing for y < ˜ y ( y ∈ D ) and strictlydecreasing for y > ˜ y ( y ∈ D ). Proposition 8.
Let d ≥ and < u ≤ and let ˆ s and F be defined as in (10)and (11). Then (a) F is unimodal with mode ˜ y = d u (1 − u ) ; (b) F has a unique zero ¯ y ∈ (0, ˜ y ) , and we have thatH d ( u ) ≥ ˆ s ( ¯ y ). Proof.
To analyse F ( y ), we define g ( s , y ) = f d ( u , s , y , ˆ x ( y ))and note that by definition F ( y ) = g ( ˆ s ( y ), y ). We haved F d y = ∂ g ∂ y | s = ˆ s ( y ) + ∂ g ∂ s | s = ˆ s ( y ) d ˆ s d y .Now ∂ g ∂ y = ∂ f d ( u , s , y , x ) ∂ y | x = ˆ x ( y ) + ∂ f d ( u , s , y , x ) ∂ x | x = ˆ x ( y ) d ˆ x d y ,where the first partial derivative is 0 by the way we defined ˆ x , and the secondone, evaluated at s = ˆ s ( y ), is 0 by (10) and the comment above it. Hence, the first partial derivative in the above formula for d F /d y is 0. Furthermore, by Lemma 7,d ˆ s ( y )/d y >
0. Thus, d F /d y has the same sign as ∂ g ∂ s | s = ˆ s ( y ) = ∂ f d ∂ s ( u , ˆ s ( y ), y , ˆ x ( y )) = − log ˆ s ( y ) + log ¡ − u − ˆ s ( y ) ¢ + log ¡ ( ˆ x ( y ) + d − ¢ .Put ˜ y = d u (1 − u ). Observe that ˆ x ( ˜ y ) = u − u (12)and hence ˆ s ( ˜ y ) = S ( ˜ y , ˆ x ( ˜ y )) = (1 − u )(1 − (1 − u ) d ). (13)Thus, after some simplifications, we find that ∂ g ∂ s | s = ˆ s ( ˜ y ) = − log s + log ¡ − u − s ¢ with respect to s is negative,d ˆ s ( y )/d y >
0, and d ˆ x ( y )/d y < ∂ g ∂ s | s = ˆ s ( y ) , andconsequently also d F ( y )/d y , takes the value 0 at y = ˜ y , and it is positive for y < ˜ y and negative for y > ˜ y . This gives the unimodality claim in (a), so we may turnto (b).We first address the existence of ¯ y . Performing some straightforward manip-ulations we see thatlim y → + F ( y ) = d −
22 ( u log u + (1 − u ) log(1 − u )) < F ( ˜ y ) = − u log u − (1 − u ) log(1 − u ) > y ∈ (0, ˜ y ) which satisfies F ( ¯ y ) =
0. By the unimodality of F , it is unique.We next show that, essentially, when computing H d , for the maximum in thedefinition of h d , we can restrict ourselves to points ( s , y ) of the form ( ˆ s ( y ), y ). Fix y < d u . We claim that for all y < d u , M d ( u , ˆ s ( y ), y ) ≤ F ( y ) = f d ( u , ˆ s ( y ), y , ˆ x ( y )). (14)To see this, note that M d ( u , ˆ s ( y ), y ) ≤ f d ( u , ˆ s ( y ), y , ˆ x ( y ))and from above ∂ f d ( u , ˆ s ( y ), y , ˆ x ( y ))/ ∂ y = log ˆ x ( y ) − log ˆ x ( y ).Since d ˆ x ( y ) /d y = d (2 u − d − d u − y ) , ˆ x ( y ) is a nonincreasing function of y .This implies that for fixed y , f d ( u , ˆ s ( y ), y , ˆ x ( y )) is maximised at y = y , whichgives (14). From this, it follows that h d ¡ u , ˆ s ( y ) ¢ = max ˆ s ( y ) ≤ y ≤ d ˆ s ( y ) M d ( u , ˆ s ( y ), y ) ≤ F ( y ). (15) SOPERIMETRIC NUMBERS OF RANDOM REGULAR GRAPHS 13
Recalling that ˆ s is a continuous increasing function of y which tends to 0 fromabove with y , it follows that if s < ˆ s ( ¯ y ), then s = ˆ s ( y ) for some y < ¯ y and h d ( u , s ) = h d ¡ u , ˆ s ( y ) ¢ ≤ F ( y ) < F ( ¯ y ) =
0, (16)where the last inequality follows since d F /d y > y ). Thus H d ( u ) ≥ ˆ s ( ¯ y ), asrequired for (b). (cid:4) Since ¯ y as in Proposition 8(b) depends on d and u , we denote it by ¯ y d , u . To beclear, let us emphasize that F ( ¯ y d , u ) = f d ( u , ˆ s ( ¯ y d , u ), ¯ y d , u , ˆ x ( ¯ y d , u )) = A d ( u ) = ˆ s ( ¯ y d , u )/ u . (17)Combining the proposition with Lemma 6, we obtain the following immedi-ately. Corollary 9.
If u → u where < u ≤ is fixed, thenI V , u ( d ) ≥ A d ( u ).By this corollary and Lemma 6(c), for any u ∈ (0, 1/2], we have i V , u ( d ) ≥ inf < u ≤ u A d ( u ). (18)Approximate values of A d ( u ) for various d and u are provided in Table 1.These were found by searching for the first zero of F = F d , u ( y ) and finding strictlypositive and negative values of F on either side of it. Recall that finding sucha zero of F as a function of y means we have found ¯ y d , u and hence A d ( u ) via(17). The entires in the table are monotonically decreasing in the columns, andthis seems likely to hold for all d . If this is true, it would follow from (18) that i V , u ( d ) ≥ A d ( u ) in general. T ABLE
1. Approximate values for A d ( u ). By Corollary 9, these are approximatelower bounds for the u -vertex expansion number I V , u ( d ). u ≈ A ( u ) ≈ A ( u ) ≈ A ( u ) ≈ A ( u ) ≈ A ( u ) ≈ A ( u ) ≈ A ( u )0.01 0.55822 1.24636 1.97397 5.71086 16.16640 30.80253 52.219310.05 0.43552 0.97129 1.52478 4.12128 9.57894 14.12199 17.140340.10 0.36513 0.80589 1.24807 3.13558 6.15315 7.78467 8.526070.15 0.31790 0.69369 1.06039 2.50085 4.35286 5.11942 5.437850.20 0.28136 0.60687 0.91620 2.04298 3.25720 3.68551 3.862670.25 0.25110 0.53536 0.79862 1.69322 2.52784 2.79584 2.908370.30 0.22503 0.47421 0.69923 1.41621 2.01058 2.19121 2.268360.35 0.20194 0.42060 0.61319 1.19112 1.62589 1.75381 1.809260.40 0.18108 0.37272 0.53737 1.00461 1.32904 1.42271 1.463830.45 0.16196 0.32936 0.46968 0.84761 1.09323 1.16336 1.194470.50 0.14420 0.28966 0.40859 0.71371 0.90142 0.95467 0.97850
5. L
OWER BOUNDS FOR THE VERTEX ISOPERIMETRIC NUMBER
For i V , u , as opposed to I V , u (for a sequence u ( n ) → u ), we must considervertex sets of cardinality less than or equal to u n . As noted in (18), the minimumof A d ( u ) over all 0 < u ≤ u gives a lower bound for i V , u . However, this turns outto be not so amenable to theoretical analysis.In this section, for the case u = i V ( d ) ≥ A d (1/2). This bound is best possible for a direct application of thefirst moment method, since it was best possible for I V , u ( d ) over all u → u = x ( y ) = s d /2 − yd − d /2 − y = y , and we get the simplified expressionˆ s ( y ) = S ( y , 1) = y (1 − d )/ d , (19)which is useful for computing F ( y ) for u = A d (1/2). Corollary 9provides us with the lower bound I V , u ( d ) ≥ A d (1/2) when u → i V ( d ) with Lemma 6, we need to also consider H d ( u )/ u for0 < u < u , we use inequality (6) to show that the case u = A d (1/2) ≤ I V , u ( d ) for any sequence u ( n ) → u ∈ (0, 1/2].Let ˆ A be the set of all ( u , s ) with s / u < A d (1/2) and 0 < u ≤ h d ( u , s ) < A . It then follows that H d ( u ) > s for all ( u , s ) ∈ ˆ A , and then,by Lemma 6, we may conclude i V ( d ) ≥ A d (1/2).To this end, define b h d ( u , s ) = max s ≤ y ≤ min{ ds , du } f d ( u , s , y , 1). (20)Since f d ( u , s , y , 1) ≥ M d ( u , s , y ), we have b h d ( u , s ) ≥ h d ( u , s ), and it suffices toshow b h d ( u , s ) < A . Our job will be made easier after we show that we have ∂ f d ( u , s , y , 1)/ ∂ y < A d (1/2), we obtain an initialestimate of the location of ˆ A with the following lemma. Define C ( d ) = ( d − d − Lemma 10.
For any d ≥ , we have < A d (1/2) < C ( d ) , and for all s < A d (1/2)/2 , b h d (1/2, s ) < .Proof. Fix d . Put y d = d ( d − d − /( d − d − a / bc = a /( bc ), that is, multiplication by juxtaposition takes precedence over ‘/’.) SOPERIMETRIC NUMBERS OF RANDOM REGULAR GRAPHS 15
We have F d ,1/2 ( y d ) = − ¡ ( d − d +
2) log 2 + log(1/( d − ¢ /2( d − + ( d −
2) log ¡ ( d − d − d − ¢ /2( d − > − ¡ ( d − d +
2) log 2 − ( d −
2) log(2 d − ¢ /2( d − = ( d − d − − ( d −
1) log 2)/2( d − = ( d −
2) log((2 d − d − )/2( d − > A d (1/2) = s ( ¯ y d ,1/2 ), where 0 < ¯ y d ,1/2 < y d . Since ˆ s is nonnegative andmonotonically increasing, we have by (19) that0 < A d (1/2) < s ( y d ) = y d (1 − d )/ d = ( d − d − = C ( d ).This establishes the first inequalities in the lemma.Since ˆ x ( y ) ≡ u = b h d (1/2, s ) = h d (1/2, s ), we have b h d (1/2, s ) < s < A d (1/2)/2 by (16), where, in this instance, ¯ y = ¯ y d ,1/2 . (cid:4) We are ready to prove the main result for this section.
Proof of Theorem 1.
Fix d ≥
3. It will be convenient to parameterize s in termsof u and a new variable r . Set s = r u and note that if r =
0, then necessarily y = ∂ V = ∂ E = b h d ( u , s ) < A . Observe that for relevant s and u , d f d ( u , s , y , 1)d y = log s d u − yd − d u − y < b h d ( u , r u ) = f d ( u , r u , r u , 1). Moreover, the final inequality in Lemma 10takes care of the case u = g d ( u , r ) : = f d ( u , r u , r u , 1) < A , where A = {( u , r ) : 0 < u < ≤ r < A d (1/2)} .For all 0 ≤ r < C ( d ), it is easy to check thatlim u → + g d ( u , r ) =
0. (21)Again, by the final inequality in Lemma 10, g d (1/2, r ) < < r < A d (1/2).Hence it suffices to show that for each 0 ≤ r < C ( d ), g d ( u , r ) is either strictlyconvex or strictly increasing in u at every 0 < u < A as follows, with a view to showing that g d ( u , r ) is either increasing or convex in each region. Define A = {( u , r ) : r =
0, 0 < u < A = {( u , r ) : 0 < r ≤ min{ c ( d ), A d (1/2)}, 0 < u < A = {( u , r ) : c ( d ) < r < A d (1/2), 0 < u < U ( r , d )} , A = {( u , r ) : c ( d ) < r < A d (1/2), U ( r , d ) ≤ u < U ( r , d ) = min n d ( d − − r )( r + d + ( d − r − r − d ( r + o and c ( d ) = ( d − d + g d in the domain corresponding to each portion of A , startingwith A . Since d g d ( u , 0)d u = ( d − u (1 − u ) > g d ( u , r ) is strictly convex in u for 0 < u < A .Assume hereafter r >
0. Note thatd g d ( u , r )d u = ζ ( r , u , d ) η ( r , u , d ) ,with ζ ( r , u , d ) = (1 − u − r u ) d + ( − + r u − r u + u − r ) d + r u ( r + η ( r , u , d ) = u (1 − u − r u )( d − d u − r u ).As u ≤ r ≤ A d (1/2) < C ( d ) <
1, we have η ( r , u , d ) >
0. Further,d ζ ( r , u , d )d u = − (1 + r )( d − d + r ) < g d ( u , r )/d u >
0, it is enough to determine that ζ ( r , 1/2, d ) = (1 − d /2) r + (1 − d /2 − d /2) r + d ( d /2 − > r and r above are both negative since d ≥
3. Therefore wehave that ζ ( r , 1/2, d ) > A since ζ ( c ( d ), 1/2, d ) = d ( d − d + > d ≥
3. So g d ( u , r ) is strictly convex in u for fixed r such that ( u , r ) ∈ A , asrequired.As seen above, ζ ( r , u , d ) is decreasing in u . Observe that whenever U ( r , d ) ≤ ζ ( r , U ( r , d ), d ) =
0. Thus for any r ∈ ( c ( d ), C ( d )), g d ( u , r ) is strictly convex in u for 0 < u < U ( r , d ). That is, g d ( u , r ) is strictly convex in u for fixed r such that( u , r ) ∈ A . On the other hand, if r ∈ ( c ( d ), C ( d )) and U ( r , d ) ≤ u < U ( r , d ) and the fact that ζ is linear in SOPERIMETRIC NUMBERS OF RANDOM REGULAR GRAPHS 17 u , we deduce that d g d ( u , r )/d u is decreasing in u . Hence, to show that g d ( u , r )is increasing in u , it suffices to show thatd g d ( u , r )d u | u = = r log((1 − r )/ r ) + r log(2 d − + d log( d /( d − r )) + log(1 − r ) = log à (1 − r ) + r (2 d − r d d r r ( d − r ) d ! > r ∈ [ c ( d ), C ( d )]. To see that the above holds for d ≥
8, observe that for thisrange of d we have r (1 − r ) − + rr < C ( d )(1 − C ( d )) − + C ( d ) c ( d ) < (2 d − µ dd − c ( d ) ¶ dC ( d ) < (2 d − µ dd − r ¶ dr .For d ≤
7, put θ d ( r ) = d g d ( u , r )/d u | u = . We haved θ d ( r )d r = − − r − + r (1 − r ) − r + d ( d − r ) < d ≥ r <
1, since each of the first three terms is less than − θ d ( c ( d )) and θ d ( C ( d )) arepositive for 3 ≤ d ≤
7. So by the concavity of θ d ( r ) in r , we have that θ d ( r ) > d ≤ r . Therefore g d ( u , r ) is strictly increasing in u for fixed r such that ( u , r ) ∈ A . Putting the above together, we conclude that g d < A ∪ A .Altogether we have shown g d < A = S i = A i . As noted earlier, this im-plies i V ( d ) ≥ A d (1/2). (cid:4) To supplement the values in Table 1, additional approximations to A d (1/2)for various d were generated by the same method, as shown in Table 2. T ABLE
2. Approximate values for A d = A d (1/2). By Theorem 1, these are ap-proximate lower bounds for the vertex isoperimetric number i V ( d ). d ≈ A d d ≈ A d d ≈ A d d ≈ A d d ≈ A d
6. A
SYMPTOTIC BOUNDS FOR i V ( d )The asymptotics of A d (1/2), as d → ∞ , can be computed as follows. For thecase u = x ( y ) = s ( y ) = y (1 − d )/ d . Hence, as discussed after the proof of Proposition 8, A d (1/2) = y (1 − d )/ d ,where ¯ y uniquely satisfies f d (1/2, 2 ¯ y (1 − d )/ d , ¯ y , 1) = f d (1/2, s , y , 1) = s log(2 d − + (log 2)/2 − s log s − (1/2 − s ) log(1/2 − s ) − ( d log 2)/2.Hence, when u = x = y does not appear in f d . Thus, we investigatethe asymptotics of s d = y (1 − d )/ d satisfying f d (1/2, s d , · , 1) = A d (1/2) > d ≥
3. Hence, for all d ≥ s d >
0. In fact, we can show that s d → d − = d log 2 + log(1 − d ) and manipulating, we obtain f d (1/2, s , · , 1) = s ¡ − log s − log 2 + log(1 − s ) + d log 2 + log(1 − d ) ¢ − ( d log 2)/2 + log 2 − log(1 − s )/2 = ( s − d log 2 − s log s + ( s − − s ) + (1 − s ) log 2 + O ( s /2 d ) = ( s − d log 2 + O (1).Setting this equal to 0, we conclude s d = + O (1/ d ) as d → ∞ . With this inmind, we make the change of variables t = − s d in the above expression, ob-taining 0 = − t d log 2 − (1/2) log(1/2) + O (1/ d ) + O (log d / d ) + (1/2) log 2and hence t = d + O (log d / d ).Therefore A d (1/2) = − d + O (log d / d ). (22)By Theorem 1 and (22), we deduce Corollary 2. Observe that Corollary 2 providesa stronger bound on the asymptotics of i V ( d ) as d → ∞ than (2).7. A NOTE ON THE EDGE ISOPERIMETRIC NUMBER
In this section we show how the above arguments can be modified to obtaina.a.s. lower bounds for the edge isoperimetric number of random regular graphs.As discussed in Section 1, Bollobás [5] computed lower bounds on i E ( d ) for all d ≥ i E ( d ), as d → ∞ ,are investigated. Moreover, it is claimed that the arguments could be modifiedfor 0 < u < i E , u ( d ) are not given, norare they given for the asymptotics of i E , u ( d ), as d → ∞ , for the cases 0 < u < i E , u ( d ) which result from directapplication of the first moment method for all d ≥ < u ≤ d → ∞ for fixed u . SOPERIMETRIC NUMBERS OF RANDOM REGULAR GRAPHS 19
For a randomly selected element of P n , d , let X ( n ) u , y , d denote the number of sub-sets of V of size un that have | ∂ E | = yn . Then E ³ X ( n ) u , y , d ´ = Ã nun !Ã d unyn !Ã d n − d unyn ! ( yn )! M ( d un − yn ) M ( d n − d un − yn ) M ( d n ) ,where M (2 m ) = (2 m )!/ m !2 m counts the number of perfect matchings of 2 m points, the binomial coefficients choose a set U consisting of un vertices and yn boundary edges, and the other factors count choices of the pairs with theobvious restrictions. Therefore, via Stirling’s approximation, ³ E X ( n ) u , y , d ´ n = ( d u ) du ( d − d u ) d − du φ ( n ) u u (1 − u ) − u y y ( d u − y ) ( du − y )/2 ( d − d u − y ) ( d − du − y )/2 d d /2 ,where φ ( n ) = n O (1/ n ) contains the factors of polynomial size before taking the n th root. Hence log E ³ X ( n ) u , y , d ´ ≤ n ( b f d ( u , y ) + o (1)), (23)where b f d ( u , y ) = d u log( d u ) + ( d − d u ) log( d − d u ) − u log u − (1 − u ) log(1 − u ) − y log y − ¡ ( d u − y ) log( d u − y ) + ( d − d u − y ) log( d − d u − y ) + d log d ¢ /2.Up to this point, these facts are essentially contained in [5]. To get lower boundson i E , u ( d ) we find where b f d < f d , defined at (6), with parameters correspondingto the edge and vertex boundary sizes and also one used to estimate a polyno-mial coefficient.)Let us collect some facts about b f d . Note that if G ∈ G n , d and | U | = un with u ≤ | ∂ E U | ≤ d un .Fix d ≥ < u ≤ b f d ( u , y ) is strictly concave in y for 0 ≤ y < d u . Indeed, so long as 0 ≤ y < d u , we haved b f d ( u , y )d y = d ( y − d u (1 − u ))2 y ( d u − y )( d − d u − y ) < b A d ( u ) = u min{ y : b f d ( u , y ) ≥ < b A d ( u ) ≤ d (1 − u ), observe thatlim y → + b f d ( u , y ) = d −
22 ( u log u + (1 − u ) log(1 − u )) < b f d ( u , d u (1 − u )) = − u log u − (1 − u ) log(1 − u ) > As we did for the case of vertex expansion, we will define a pointwise measureof edge expansion. For a sequence u = u ( n ) with 0 < u ≤ n , we definethe u-edge expansion number to be I E , u ( d ) = sup ½ ℓ : min U ⊂ V , | U |= un | ∂ E U | un ≥ ℓ a.a.s. in G n , d ¾ .We state here analogues of Lemmas 5 and 6 for the case of edge expansion.We do not provide the proofs since they are very similar. Lemma 11.
Fix < u ≤ . Theni E , u ( d ) ≥ inf ≤ u ≤ u inf w → u I E , w ( d ), where the second infimum is over sequences w ( n ) with < w ≤ .Proof. The proof is analogous to that of Lemma 5. (cid:4)
Lemma 12. b A d has the following properties. (a) Fix < u ≤ . If u = u ( n ) → u as n → ∞ , thenI E , u ( d ) ≥ b A d ( u ). In the case that u → + , I E , u ( d ) ≥ d − For any < u ≤ , we havei E , u ( d ) ≥ inf < u ≤ u b A d ( u ). Proof.
The proof is analogous to that of Lemma 6. (Note that the small sets prop-erty, as discussed at (3) and (4), also holds for edge expansion.) (cid:4)
As we now prove, b A d ( u ) is, in fact, a lower bound for i E , u ( d ) for all d ≥ < u ≤ Proof of Theorem 3.
Fix d ≥ < u ≤ < w ≤ u and parameterize the variable y as r w , where r is a new variable. Put B = © ( r , w ) : 0 ≤ r < b A d ( u ), 0 < w ≤ u ª .Once we show g d ( r , w ) = b f d ( w , r w ) < B , the theorem will follow by apply-ing Lemma 12 and inequality (23). Partition B as follows: B = B ∩ © ( r , w ) : w ≤ W r , d or r ≤ R w , d ª , B = B \ B ,where W r , d = d ( d − − r )( d − d + r ) , R w , d = d ( d − − w ) d + ( d − w . SOPERIMETRIC NUMBERS OF RANDOM REGULAR GRAPHS 21
We have d g d ( r , w )d w = η ( d , r , w ) ζ ( d , r , w ) ,where η ( d , r , w ) = (1 − w ) d − (2(1 − w ) + r (1 + w )) d + r w , ζ ( d , r , w ) = w ( d − d w − r w )(1 − w ).As w ≤ u ≤ r < d u , ζ ( d , r , w ) >
0. Also, we have η ( d , r , W r , d ) = η ( d , R w , d , w ) = w < W r , d or r < R w , d , then η ( d , r , w ) >
0. Hence, over B , g d ( r , w ) is con-vex in w for any fixed r .Regarding B , we know that d g d ( r , w )d w < g d ( r , w )/d w is decreasing in w . Thus, to show that g d ( r , w ) for a fixed r is increasing in w over B , it suffices to show that θ u ( r ) = d g d ( r , w )d w | w = u > R u , d ≤ r ≤ d (1 − u ), since R w , d is decreasing in w and b A d , u < d (1 − u ). First,observe that d θ u ( r )d r = d (1 − u ) r − d (1 − u + u )2 r ( d − r )( d − d u − r u ) < < r < d (1 − u + u )/(1 − u ). So as2 d (1 − u + u )(1 − u ) − d (1 − u ) = d (1 − u )(2 u − u + − u > θ u is strictly concave in r over the interval in question. Thus we check that θ u ispositive at the endpoints. The right endpoint is positive since, after some simplemanipulations, we see thatd θ u ( d (1 − u ))d u = − u (1 − u ) < θ ( d (1 − = θ u ( R u , d ) = ( d −
1) log ³ u − u ´ + R u , d ϕ u ( d ) + d ψ u ( d ),where ϕ u ( d ) = ( d − d u − R u , d u )( d u − R u , d u )( R u , d u ) = d (( d − u + u (1 − u )( d − , ψ u ( d ) = d − d u − R u , d ud u − R u , d u = d (1 − u )2 u (( d − u +
1) .
Hence, after some simplifications, we find thatd θ u ( R u , d )d d = δ u ( d ) log ϕ u ( d ) + γ u ( d ) d ( d − d − u + d − u + d ) ,where δ u ( d ) = − d u (1 − u )( d − d − u + γ u ( d ) = (( d − u + d ) ¡ (4 u − u − d − u − u + d − u (1 − u ) ¢ .Note that the coefficients of the second term in γ u ( d ) are nonpositive for all 0 < u ≤ δ u ( d ) < γ u ( d ) < d ≥ < u ≤ ϕ u ( d )d d = − + u ) d + − u )) u (1 − u )( d − < d →∞ ϕ u ( d ) = − u ≥ ϕ u ( d ) ≥ d ≥ < u ≤ θ u ( R u , d )/d d is decreasing in d for any fixed 0 < u ≤ d →∞ d θ u ( R u , d )d d = − log(1 − u ) + u log 21 + u > < u ≤ θ u ( R u , d ) is increasing in d , and thus, since θ u ( R u ,3 ) ≥ θ ( R ) ≈ < u ≤ θ u ( R u , d ) ≥ d and u . Alto-gether, g d ( r , w ) is increasing in w over B for any fixed r .Finally, for any 0 ≤ r ≤ d (1 − u ), it is easily seen thatlim w → + g d ( r , w ) = i E , u ( d ) ≥ b A d ( u ). (cid:4) Approximate values for b A d ( u ) are listed in Table 3. T ABLE
3. Approximate values for b A d ( u ). By Theorem 3, these are approximatelower bounds for the u -edge isoperimetric number i E , u ( d ). u ≈ b A ( u ) ≈ b A ( u ) ≈ b A ( u ) ≈ b A ( u ) ≈ b A ( u ) ≈ b A ( u ) ≈ b A ( u )0.01 0.57080 1.29152 2.07102 6.31585 20.00259 43.58306 91.532590.05 0.46150 1.06879 1.73912 5.49362 17.96765 39.83142 84.744260.10 0.39850 0.93300 1.52904 4.91775 16.36950 36.65008 78.569940.15 0.35544 0.83739 1.37806 4.48034 15.07700 33.96870 73.178300.20 0.32140 0.76038 1.25487 4.11019 13.93791 31.54266 68.189080.25 0.29262 0.69435 1.14821 3.78107 12.89392 29.27612 63.453610.30 0.26728 0.63557 1.05254 3.47967 11.91538 27.12097 58.895260.35 0.24435 0.58192 0.96469 3.19830 10.98467 25.04690 54.467940.40 0.22318 0.53205 0.88263 2.93177 10.08979 23.03451 50.139460.45 0.20332 0.48501 0.80492 2.67658 9.22247 21.06947 45.887310.50 0.18447 0.44011 0.73051 2.43002 8.37615 19.14025 41.69360 SOPERIMETRIC NUMBERS OF RANDOM REGULAR GRAPHS 23
With Theorem 3 in hand, we can derive lower bounds on the asymptotics of i E , u ( d ) as d → ∞ . Proof of Corollary 4.
Put ψ ( u ) = − u ) q log( u − u (1 − u ) u − ).For any c > b f d ( u , u ( d (1 − u ) − c p d )) − log( u − u (1 − u ) u − ) = cu p d µ d ( u , c ) − d ν d ( u , c ),where µ d ( u , c ) = ( d u (1 − u ) − cu p d ) ( d u + cu p d )( d (1 − u ) + cu p d ) , ν d ( u , c ) = µ − c (1 − u ) p d ¶ u (1 − u ) µ + cu p d ¶ u µ + cu (1 − u ) p d ¶ (1 − u ) .Observe that lim d →∞ cu p d µ d ( u , c ) = − c − u ) and lim d →∞ d ν d ( u , c ) = − c − u ) .Hence lim d →∞ b f d ( u , u ( d (1 − u ) − c p d )) = − − u ) ( c − ψ ( u ))( c + ψ ( u )),and thus, for any ǫ > d , b A d ( u ) ≥ d (1 − u ) − ( ψ ( u ) + ǫ ) p d .Applying Theorem 3, the result is obtained. (cid:4) Note that in the case u = CKNOWLEDGMENTS
BK was supported by NSERC of Canada. NW carried out this research at theDepartment of Combinatorics & Optimization, University of Waterloo, and wassupported by the Canada Research Chairs program and NSERC of Canada. Thisproject began during a visit by BK to the University of Waterloo. BK would liketo thank NW for support and hospitality during this time. R EFERENCES [1] A
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RETT K OLESNIK D EPARTMENT OF M ATHEMATICS , U
NIVERSITY OF B RITISH C OLUMBIA
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