A Survey of Ramanujan Graphs

Wen-Ch’ing Winnie Li∗ April 29, 2014

Abstract In this article we give an overview of the development of Ramanu- jan graphs. We explain how the subject started, the known explicit constructions of such graphs, the analogy between the spectral analysis of graphs and Riemannian manifolds, the distribution of the spectra of regular graphs, and an application from graph theory to modular forms.

1991 Mathematics Subject Cassiﬁcation: 05C75, 11F11, 11T23

1 Introduction

Ramanujan graphs are regular graphs with small nontrivial eigenvalues. They are close to regular random graphs, and hence ”expand” well. This and other features make them useful in communication networks. They also have applications in many other areas, including extremal graph theory and computa- tional complexity, see, for instance, [Cg] and [FW]. In particular, Ramanujan graphs can be used to construct low density parity check codes (cf [M3]). The focus of this article, however, is not on the aforementioned applications; instead, we shall take a more graph-theoretical viewpoint, following the historical development of the subject. Our emphasis is on the rich interplay between two seemingly far apart subjects, such as the mutual applications between Ramanujan graphs and number theory, as explained in sections 3 and 6, and the analogies of spectral analysis on graphs and Riemannian

∗Research supported in part by NSF grant no. RII90-03126, NSA grant no. MDA904- 92-H-3054 and a grant from National Science Council in Taiwan

1 manifolds, as discussed in section 4. This kind of intimate relationship has fruitfully inspired the advancement of the study within each individual field. The paper is organized as follows. Starting with the fundamental problem of constructing efficient communication networks at a cost not exceeding a fixed amount, in section 2 we explain mathematicians’ two approaches to solve this problem: one uses Kazhdan’s property T , and the other uses the spectra of graphs. At present, the latter method appears to be more desirable because we can estimate the efficiency of a regular graph via its largest nontrivial eigenvalue. From the growth of the largest nontrivial eigenvalue (Theorem 1), we are naturally led to the definition of Ramanujan graphs. As discussed in section 3, to date, there are three known explicit constructions of Ramanujan graphs, they are based on quaternion groups, finite abelian groups, and finite nonabelian groups, respectively. These constructions rely on deep number-theoretic results: the estimate of eigenvalues of Hecke operators on classical weight 2 cusp forms (the former Ramanujan-Petersson conjecture, established by Deligne as a consequence of the Weil conjectures), certain character sum estimates resulting from the Riemann hypothesis for curves over finite fields proved by Weil, and representation theory for GL2 over finite fields and character sum estimates, respectively. Graphs may be regarded as discrete Riemannian manifolds. And indeed, one finds many results on both subjects which are alike. In section 4 we exhibit some spectral analogies, especially the connection between the Riemann hypothesis for the attached zeta functions and the spectra of their universal covers. The distribution of eigenvalues of regular graphs is considered in section 5. We discuss this from three aspects: concentration of large eigenvalues, approxi- mated eigenvalue distribution of large graphs, and overall eigenvalue distribution for graphs with known girth. Finally in section 6 we apply the result on eigenvalue distribution (Theorem 9) to a family of (Ramanujan) graphs constructed in section 3 using quaternion groups to obtain the asymptotic behaviour of the dimension of cusp forms of weight 2 for certain congruence subgroups of SL2(Z) which are eigenfunctions of a fixed Hecke operator Tp with integral eigenvalues.

2 Background

A fundamental problem in communication network is to construct eﬃcient networks at a cost not exceeding a ﬁxed amount. To phrase this mathemati-

2 cally, represent a network by a graph G. The efficiency may be measured by the so-called magnifying constant c of the graph defined by |∂X| c = min , X |X| where X runs through all subsets of vertices of G of size |X| not exceeding half of the size of G, and ∂X, called the boundary of X, denotes the set of vertices in G adjacent to X but not in X. Here an upper bound on the size of X is set, for as X gets large, the ratio of the size of the boundary of X versus the size of X becomes small and it does not reveal how ”expanding” the smaller subsets of G are. The problem then is to find graphs G with fixed number of vertices and large magnifying constant c while the degree (that is, the number of edges) at each vertex remains bounded, say, at most k. Or even better, get an infinite family of graphs with the same k and with (good) c uniformly bounded from below. Probabilistically speaking, if a graph is random, then its magnifying constant is usually good; on the other hand, to determine a given graph’s magnifying constant is a very difficult task in general. Therefore it is desirable to have explicit constructions of graphs with a lower bound of the magnifying constant readily computable from construction. For this, two systematic methods emerged in the past two decades, which we briefly describe below. The first method uses Kazhdan’s property T . A locally compact group Γ is said to have property T if the trivial representation is isolated from other unitary representations of Γ. To measure how far apart the trivial representation is from the other representations, there is a so-called Kazhdan’s constant κ. The topology on the representations is similar to the ”compact- open” topology with the role of compact sets played by generators of the group Γ. Thus the distance κ from the trivial representation to other representations depends on the choice of generators. The idea of relating the magnifying constant c of a graph to Kazhdan’s constant κ was due to Margulis, who in 1975 took Γ to be the semidirect 2 product of SL2(Z) by Z and used it to construct an infinite family of bipar- κ2 tite graphs with degree k = 5 and magnifying constant c ≥ 2 , where κ is Kazhdan’s constant arising from the four standard generators of Γ [M1]. Un- fortunately, he was not able to compute κ. In 1981, Gabber and Galil [GG] followed the same method but took a different set of generators to construct a new family of 5–regular graphs, and they succeeded in computing the magnifying constant explicitly. Since then, there were similar constructions with

3 improved magnifying constants. The reader is referred to the review article by Bien [B] for more details on this method. There are two drawbacks in this approach: the ﬁrst is that there is no known explicit lower bound of c in terms of κ; the second is that it is diﬃcult to compute κ. The second method uses the largest nontrivial eigenvalue of a regular graph. Given a graph G, denote by A = A(G) its adjacency matrix. More precisely, the rows and the columns of A are parametrized by the vertices of G, and the vv0-th entry of A is the number of edges from the vertex v to the vertex v0. (If G is undirected, the edges are viewed as directed both ways.) We may regard A as an operator on the space of functions on (vertices of) G, which sends a function f to Af whose value at a vertex x is given by X (Af)(x) = f(y), y where y runs through all out-neighbors of x. The eigenvalues of A are called the spectrum of G. Assume that G is k–regular, connected, and undirected. Then the eigenvalues of G are real and satisfy

k = λ1 > λ2 ≥ · · · ≥ λ|G| ≥ −k.

The eigenvalues of a k–regular (directed or undirected) graph G with absolute value k are called trivial eigenvalues, and the remaining ones called nontrivial eigenvalues. In 1984 Tanner [Ta] showed that the magnifying constant c of G has a lower bound in terms of the largest nontrivial eigenvalue λ2 as follows: k c ≥ 1 − . 3k − 2λ2

Conversely, in 1985 Alon and Milman [AM] gave an upper bound of λ2 in terms of c :

c2 k − λ ≥ . 2 4 + 2c2

From these two inequalities one sees immediately that the smaller λ2 is, the larger c is. Hence our problem is reduced to ﬁnding regular graphs with small nontrivial eigenvalues. For this purpose, deﬁne, for a regular graph G,

4 λ = λ(G) = max |λi|, λi

where λi runs through all nontrivial eigenvalues of G. Clearly,

k > λ ≥ λ2. The number λ also gives other information of the graph. For instance, in 1989 Chung [Cg] obtained the following upper bound of the diameter of a k–regular undirected graph G with n vertices in terms of λ(G):

log (n − 1) diam G ≤ . log k/λ(G) When G represents a communication network, its diameter measures transmission delay. Therefore this gives another good reason for wanting to construct regular graphs with small λ. The reader is referred to the survey paper [Mo] by Mohar and the papers therein for connections between eigenvalues and other properties of graphs. In [DS] Delorme and Sol´edis- cussed Chung’s diameter bound for bipartite graphs and relations between eigenvalues and certain parameters of linear codes. How small can λ be? The well-known result of Alon and Boppana asserts that Theorem 1. ([LPS]) For k–regular undirected graphs G, we have √ lim inf λ(G) ≥ 2 k − 1 as the size of G tends to inﬁnity. This bound can also be derived from the more recent result of Nilli which gives a lower bound of λ2 in terms of the diameter of the graph: Theorem 2. ([N]) Let G be a k–regular undirected graph. If the diameter of G is ≥ 2l + 2 ≥ 4, then √ √ 2 k − 1 − 1 λ(G) ≥ λ (G) > 2 k − 1 − . 2 l

We remark that Theorem 1 also holds for k–regular directed graphs G with A(G) diagonalizable by unitary matrices. This leads to the following deﬁnition of Ramanujan graphs, which, loosely speaking, are regular graphs with small eigenvalues.

5 Definition. A graph G is called a Ramanujan graph if (a) G is k–regular,√ (b) λ(G) ≤ 2 k − 1, (c) A(G) is diagonalizable by unitary matrices. Here the third condition is automatically satisfied if G is undirected, for its adjacency matrix is then symmetric and hence diagonalizable by an orthogonal matrix. This definition first appeared in [LPS] for undirected graphs, and was later extended to include directed graphs in [L1]. As explained in [L1], directed Ramanujan graphs also give rise to undirected Ramanujan graphs.

3 Explicit constructions of Ramanujan graphs

To date, there are three systematic ways to construct Ramanujan graphs explicitly, they are all number-theoretic. We shall survey these methods chronologically. (I) Ramanujan graphs based on adelic quaternionic groups Graphs constructed using this method will have valency k = p + 1, where p is a prime. The general method is as follows. Take a definite quaternion algebra H defined over Q unramified at p and ramified at ∞. Denote by D the multiplicative group H× divided by its center. Let X be the double coset space of the adelic points of D : Y X = D(Q)\D(AQ)/D(R) D(Zq), q prime

where Zq is the ring of integers of the q-adic ﬁeld Qq. By the strong ap- proximation theorem, the above (global) double coset space can be expressed locally :

1 X = D Z[ p ] \D(Qp)/D(Zp) 1 = D Z[ p ] \PGL2(Qp)/PGL2(Zp).

Here D(Qp) is isomorphic to P GL2(Qp) since H is unramified at p. The right coset space P GL2(Qp)/P GL2(Zp) has a natural structure as a (p + 1)- 1 regular infinite tree T (see [Se1]), and the discrete group Γ = D Z[ p ] acts on T . Since H is ramified at ∞, Γ\T is a finite (p + 1)-regular graph, which is the graph structure on X. We may replace Γ by a congruence subgroup Γ0 and thus obtain a finite cover X0 of X.

6 The functions on X are certain automorphic forms on the adelic quaternion group D(AQ), including the constant functions, which are eigenfunctions of the adjacency matrix A(X) with eigenvalue p + 1. By the representation theory developed by Jacquet and Langlands [JL], the orthogonal complement of the constant functions can be identified with certain classical cusp forms of weight 2. When regarded as such, the operator A(X) is nothing but the Hecke operator Tp acting on cusp forms. The graph X is Ramanujan because Deligne [D2, D3] has shown that an eigenvalue λ of Tp on cusp forms of weight 2 satisfies √ √ |λ| ≤ 2 p = 2 k − 1, the former Ramanujan-Petersson conjecture. Margulis [M2] and, independently, Lubotzky, Phillips and Sarnak [LPS] (see also [Cu]) took H to be the Hamiltonian quaternion, they obtained an infinite family of (p+1)-regular Ramanujan graphs by taking congruence subgroups of Γ. On the other hand, Mestre and Oesterlé[Me] chose H to be H`, the quaternion algebra over Q ramified only at ∞ and at a prime ` 6= p, and constructed the graph X` as above by taking the maximal congruence subgroup. As ` tends to infinity, they obtained an infinite family of Ramanujan graphs. In general, one may both vary quaternion algebras and take congruence subgroups to construct infinitely many Ramanujan graphs. The same argument works when the base field Q is replaced by a function field of one variable over a finite field. In that case the resulting graphs are Ramanujan because Drinfeld [Dr] has proved the Ramanujan conjecture for GL2 over function fields. This is done in Morgenstern’s thesis. Note that the graphs so constructed have valency k = q + 1 with q a power of a prime. In [P] Pizer constructed (p+1)-regular Ramanujan graphs allowing multiple edges by using the action of the classical Hecke operator at p on spaces of certain theta series of weight 2. (II) Ramanujan graphs based on finite abelian groups We start by giving a general recipe. Let G be a finite abelian group and S a k-element subset of G. Define two k-regular graphs on G, called the sum graph Xs(G, S) and the difference graph Xd(G, S) as follows : the out-neighbors of x ∈ G in the sum graph are elements in {y ∈ G : x + y ∈ S} = −x + S, and those in the difference graph are in {y ∈ G : y − x ∈ S} = x+S, respectively. The graphs Xs and Xd so constructed enjoy the following

7 properties: (d) Their adjacency matrices are diagonalizable by unitary matrices; (e) Their eigenvalues in absolute value are the same, given explicitly by

X ψ(s) for characters ψ of G.

s∈S Hence if we can ﬁnd a suitable group G and a subset S such that

X √ ψ(s) ≤ 2 k − 1

s∈S for all nontrivial characters ψ of G, then we will have constructed two Ramanujan graphs Xs(G, S) and Xd(G, S). This observation converts a combinatorial problem to a number-theoretic problem. This method was first used by Chung [Cg] for certain cyclic groups, the general setting as above was formulated by Li [L1]. We state a few character sum estimates which led to the construction of Ramanujan graphs. In what follows, F denotes a finite field with q elements, and Fn denotes a degree n field extension of F. Let Nn be the set of elements in Fn with norm to F equal to 1. In 1977 Deligne published the following estimate of generalized Klooster- man sum : Theorem 3. ([D1]) For all nontrivial characters ψ of Fn, we have

X n−1 ψ(s) ≤ n q 2 .

s∈Nn

When n = 2,N2 has cardinality k = q + 1. The above theorem implies that Xs(F2,N2) and Xd(F2,N2) are Ramanujan graphs. Deligne proved the above theorem using étalecohomology and algebraic geometry. We exhibit more character sum estimates below using the Riemann hypothesis for curves over finite fields which was proved by Weil [W1]. The reader is referred to [L1] for more details. Let t be an element in Fn such that Fn = F (t). Assume n ≥ 2. Let tq − a S = : a ∈ F ∪ {∞} . n t − a

8 When a = ∞, the quotient is understood to be 1. Note that tq − a is the q image of t − a under the Frobenius automorphism x 7→ x , hence the set Sn is contained in Nn. Theorem 4.([L1]) For all nontrivial characters χ of Nn, we have

X √ χ(s) ≤ (n − 2) q.

s∈Sn

The set Sn has cardinality k = q + 1. Thus we get interesting Ramanujan graphs by taking G = Nn for n = 3, 4 and S = Sn. Theorem 5.([L1]) For all nontrivial characters ψ of Fn, we have

X √ ψ(s) ≤ (2n − 2) q.

s∈Sn

Hence we obtain Ramanujan graphs by taking G = F2 and S = Sn. Remark. When n = 2, we find S2 = N2, hence the above statement yields immediately that Deligne’s theorem (Theorem 3) for n = 2 follows from the Riemann hypothesis for curves over finite fields. Theorem 6.([L1]) For all nontrivial characters (χ, ψ) of Nn × Fn, we have X √ χ(s) ψ(s) ≤ (2n − 2) q.

s∈Sn

Again we get Ramanujan graphs by taking G = N2 × F2, and S the diagonal imbedding of S2 in N2 × F2. As noted before, S = N2 when n = 2, thus we get Corollary 1. For all nontrivial characters (χ, ψ) of N2 × F2, we have

X √ χ(s) ψ(s) ≤ 2 q.

s∈N2 In view of the analogous character sum estimate for split degree 2 algebra extension of F proved by Mordell, Deligne in [D1] conjectured that the same bound should hold for twisted generalized Kloosterman sums, namely, Conjecture (Deligne) For all nontrivial characters (χ, ψ) of Nn × Fn, we have X n−1 χ(s) ψ(s) ≤ nq 2 .

s∈Nn

9 Corollary 1 above establishes Deligne’s conjecture for the case n = 2. In fact, the conjecture in this case follows from the Riemann hypothesis for curves over finite fields. Using the same method, one can re-prove several other known character sum estimates, leading to the construction of Ramanujan graphs along the same line (cf. [L1]). (III) Ramanujan graphs based on finite nonabelian groups The Ramanujan graphs constructed by this method are based on finite nonabelian groups, which have some features in common with the previous two constructions. As before, we start with a general approach. Let G be a finite group and let K be a subgroup of G. For a K–double coset S, denote by AS the operator on the space of functions on the coset space G/K which sends a function f to another function ASf given by X (ASf)(xK) = f(xyK), x ∈ G. y∈S/K

In other words, AS is the adjacency matrix of the Cayley graph

XS = Cay(G/K, S/K) whose vertices are the K–cosets of G, and the outneighbors of xK are xyK as y runs through all coset representatives of S/K. Clearly, XS is k– |S| regular with k = |K| . In order to determine if XS is a Ramanujan graph, we need to estimate its eigenvalues. For this purpose, we make the following assumption: −1 (f) For each K–double coset S, we have S = S (so that XS is undirected), and the operators AS are mutually commutative for all K−double cosets S. Then, as shown in [L2], the eigenvalues of XS are parametrized by certain characters of the group G. More precisely, we have Theorem 7.([L2]) Assume (f). Then for any K–double coset S of G, the eigenvalues of the graph XS are

|S| X λ = tr π(ks) S,π |K|2 k∈K (with multiplicities), where S = KsK, and π runs through all irreducible representations of G with nonzero K–invariant vectors.

10 This is the nonabelian analogue of (e). Indeed, if G were abelian and S were a finite union of K–double cosets, then all irreducible representations of G would be one-dimensional, and the formula above would reduce to the formula in (e) (with the group being the quotient group G/K). A. Terras and her students constructed Ramanujan graphs using this method [ACPTTV], [CPTTV]. More specifically, they took the group G to be GL2(F ), where F is a finite field with q elements, q√odd. Let δ be an element in F which is not a square so that the field F ( δ) is a quadratic extension F2 of F . Its multiplicative group can be imbedded in G, which we choose to be

a bδ K = ∈ G : a, b ∈ F . b a The coset space G/K can be identified with the ”finite analogue” of the Poincaréupper half-plane:

y x H = : y ∈ F × and x ∈ F . 0 1 Let S be any K–double coset with |S| > |K|, then |S/K| = q + 1 and such double cosets are parametrized by a ∈ F, a 6= 0, 4δ. More precisely, y x there is an a as described such that S = S K, where (x, y) runs 0 1 through all the F -points of the ellipse x2 = ay + δ(y − 1)2. The Cayley graph XS is (q + 1)–regular. To show that it is Ramanujan, we have to analyze its eigenvalues. There are two types of nontrivial irreducible representations of G with nonzero K–invariant vectors: (i) The ﬁrst type arises from the nontrivial characters χ of F ×. For such a χ, deﬁne the function f on G/K by

y x f K = χ(y). 0 1

The left translations of f by G yields an irreducible representation πχ with X λS,πχ = λS,χ = χ(y). y∈F ay+δ(y−1)2=x2

11 × (ii) The second type arises from inducing characters ω of F2 which are nontrivial on N2, the kernel of the norm map from F2 to F . Denote the representation so obtained by πω. The eigenvalue λS,πω = λS,ω was computed by Soto-Andrade [SA] :

X a λS,ω = ε − 2 + 2x ω(z), √ δ z=x+y δ∈N2 x,y∈F where ε is the function on F equal to 1 on squares in F ×, −1 on nonsquares in F × and 0 at 0. Using Weil’s estimate [W2], R. Evans and H. Stark independently showed √ that |λS,χ| ≤ 2 q; while using étalecohomology and algebraic geometry, N. Katz [K] showed that λS,ω satisfied the same bound. It was proved in [L3, chap 9, Prop. 3] that both inequalities followed from the Riemann hypothesis for curves over finite fields. Therefore XS is a Ramanujan graph.

4 Comparisons

Finite undirected regular graphs bear a lot of resemblance with closed Rie- mannian manifolds, and there are parallel results and phenomena. In fact, some results were proved for manifolds first. We exhibit some analogies below. Let G be a finite connected k–regular undirected graph and let M be a closed connected Riemannian manifold of dimension n. (1) Laplacian operator ∆ Denote by C0(G) the space of functions on vertices of G, and by C1(G) the space of functions on edges of G. Define a linear map d from C0(G) to C1(G) by

(df)(e) = f(e+) − f(e−) for any edge e of G. Here we have chosen an orientation on each edge e with e+ (resp. e−) being the ending (resp. starting) vertex of e. Denote the adjoint of d by d∗, which is a linear function going in opposite direction. In our case, d∗ is also the transpose of d. The Laplacian operator ∆ is deﬁned to be d∗d. An easy computation shows that if we choose as a basis of C0(G) the characteristic functions of the vertices of G, then ∆ is representated by the matrix D − A, where A = A(G) is the adjacency matrix of G, and D is the degree matrix of G. Since G is k–regular, D is k times the identity

12 matrix. The spectrum of ∆ is thus k minus the eigenvalues of G, and hence is a sequence of real numbers between 0 and 2k. On M the Laplacian operator ∆ is deﬁned the same way with C0(M) and C1(M) being the spaces of smooth functions and 1-forms on M, respectively. It has nonnegative eigenvalues:

0 = λ0(M) < λ1(M) ≤ λ2(M) ≤ · · · .

In particular, λ1(M) plays the same role as k − λ2(G). (2) The constant c Corresponding to the magnifying constant c = c(G) on G is the Cheeger constant c = c(M) on M deﬁned by

area(S) c(M) = inf . S min(vol(M1), vol(M2)) Here S runs through (n−1)–dimensional submanifolds of M which divides M into two parts M1 and M2 with at least one part having ﬁnite volume (so that S is the common boundary of M1 and M2), and the ”area form” on S is induced from the volume form on M. In 1970 Cheeger [Cr] showed that

c(M)2 λ (M) ≥ , 1 4

which is similar to the lower bound for λ2(G) alluded to in section 2. (3) Universal cover and its spectrum The universal cover of G is a k–regular infinite tree T . In the event that q = k − 1 is a prime power, there is a local field K with q elements in its residue field such that T is isomorphic to the natural tree structure on the coset space GL (K)/GL (O )Z(K), where Z denotes the center of GL , and 2 2 K √ √2 OK denotes the ring of integers of K. The spectrum of T is [−2 q, 2 q]. Assume that M is a closed Riemann surface with a metric of constant curvature -1. Its universal cover is the Poincaréupper half-plane, which can be identified with the coset space GL2(R)/O2(R)Z(R). The spectrum of the 1 universal cover is [ 4 , ∞). (4) Zeta function and its singularities Attached to the graph G is the Ihara zeta function in one complex variable s defined by

13 Y Z(s) = (1 − q−s·`(p))−1. p Here p runs through prime geodesic cycles in G, that is, cycles with no backtracking, no repetitions, and regardless of their initial vertices, and `(p) denotes the length of p. It is known that Z(s) converges absolutely for ~~1, and in fact it is a rational function in u = q−s (cf. [I], [Su1], [Su2]):~~

~~Z(s) = (1 − u2)−g · det(I − Au + qu2)−1. q−1 The constant g in the formula above is 2 |G|. Attached to the manifold M is the Selberg zeta function, also in one complex variable s, given by~~

~~∞ Y Y Z(s) = (1 − e−(s+m)`(p)). m=0 p Here, as before, p runs through all prime closed geodesics in M and `(p) denotes the length of p. The zeros of Z(s) in the vertical strip 0 ≤~~

~~5 Distribution of eigenvalues~~

~~To an undirected k–regular graph G we attach a measure µG deﬁned by the eigenvalues of G as follows:~~

~~1 X µ = δ √ , G |G| λi/ q λi~~

14 where λi runs through all eigenvalues of G, δx is the Dirac measure supported at the point x, and q = k − 1. The measure µ is supported on the √ G √1 interval [−M,M], where M = q + q . More precisely, for a test function f deﬁned on R, we have

Z M 1 X √ µG(f) = f(x) µG(x) = f(λi/ q). −M |G| λi In particular, if f is the characteristic function of the interval [a, b], then µG(f) gives the proportion of eigenvalues (with multiplicity) of G falling in [a, b]. The following result is a reﬁnement of Theorem 1, which deals with the distribution of large eigenvalues of regular graphs. Theorem 8.(Serre [Se2], Lubotzky-Phillips-Sarnak [LPS], Feng-Li [FL]) Given a positive number , there exists an eﬀectively computable constant c, depending only on and k, such that for all k–regular graphs G, we have

Z M µG(x) ≥ c. 2− A primitive cycle in a graph is a (connected) cycle without backtracking. For each positive integer l denote by cl(G) the number of primitive cycles in G of length l. McKay studied the overall expected distribution of eigenvalues of a large regular graph. His result is Theorem 9.([Mc]) Let {Gm} be a family of connected undirected k– regular graphs with |Gm| → ∞ as m → ∞. Assume (∗) For each integer l ≥ 1, cl(Gm) |Gm| → 0 as m → ∞.

Then the sequence of measures {µGm } converges weakly to the measure µ supported on [−2, 2] given by r k 1 x2 µ = · 1 − dx M 2 − x2 π 4 √ √1 with M = q + q , and dx being the Lebesque measure on R. Notice that µ is the normalized measure on the universal cover, the k– regular inﬁnite tree. In particular, it is a continuous measure, hence each point on the real line has measure zero. This yields immediately the following Corollary 2. Let {Gm} be as in Theorem 9 satisfying (∗). For each real number α, the multiplicity of α as an eigenvalue of Gm is o(|Gm|) as m → ∞.

15 Remark. The condition (∗) is a mild assumption since McKay showed in [Mc] that it holds on average. Let G be an undirected k–regular graph with girth g. The theorem above also gives some information on the distribution of eigenvalues of G. More precisely, we have µG(pn) = µ(pn) for all polynomials of degree n < g. Choose a family of polynomials {pn(x)} orthogonal with respect to the measure µ. The roots of pn(x) are g−1 real, distinct, and lie in [−2, 2]. Put u = [ 2 ]. Arrange the roots of pu(x) in increasing order: −2 < r1 < r2 < ··· < ru < 2. It follows from the general property of orthogonal polynomials that between any two√ consecutive roots ri, ri+1 of pu there lies some normalized eigenvalue λj/ k − 1 of G. Soléin [So] showed that there is a normalized eigenvalue√ of G less than r1. Clearly, G has a normalized eigenvalue, namely, k/ k − 1, which is greater than ru. To obtain a nontrivial estimate of the second largest eigenvalue, use the modified measure ν = (M −x)µ and choose a family of polynomials {qn(x)} orthogonal with respect to ν. Then one can show that the normalized second largest eigenvalue of G is greater than the g−2 largest root of qv(x), where v = [ 2 ]. Details will appear in a forthcoming joint work with Solé. The results in this section are generalized to hypergraphs in [FL].

~~6 An application to modular forms~~

We conclude this article by exhibiting an application of Theorem 9 to modular forms. Fix a prime p. For a positive integer N prime to p, denote by 0 C Γ0(N), 2 the space generated by cusp forms of weight 2 for Γ0(N) which are eigenfunctions of the Hecke operator Tp with integer eigenvalues. Theorem 10. (i) (Serre [Se2]) Let {li} be a sequence of primes diﬀerent from p such that li → ∞ as i → ∞. Then for i large, we have 0 dim C Γ0(li), 2 = o(li).

(ii) (Li [FL]) Let {Mi} be a sequence of positive integers prime to p such that Mi = liNi with li a prime not dividing Ni and li → ∞ as i → ∞. Then for i large, we have 0 dim C Γ0(Mi), 2 = o(Mi)

~~16 0 if the Mi s have a bounded number of prime factors; otherwise we have 0 dim C Γ0(Mi), 2 = o(Mi ln ln Mi).~~

To prove this theorem, consider the family of graphs {X`} constructed from quaternion algebras H` ramified only at ∞ and ` 6= p, as discussed in section 3. Then one shows that this family of graphs satisfies the condition (∗) because for each given n, the number of (equivalence classes of) supersingular elliptic curves in characteristic ` which are self-isogenous by a cyclic group of order pn is bounded by a number independent of `. The first statement follows from Corollary 2. To prove the second statement, apply the same argument to finite covers of X` obtained from choosing appropriate congruence subgroups in the adelic setting. See [FL] for more details.

~~References~~

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