A Survey of Ramanujan Graphs

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 op- erators 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 j@Xj c = min ; X jXj where X runs through all subsets of vertices of G of size jXj 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 ≥ · · · ≥ λjGj ≥ −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 jλij; 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 Solédis- 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 p lim inf λ(G) ≥ 2 k − 1 as the size of G tends to infinity.

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