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14.10 Ramanujan and Expander Graphs

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14.10 Ramanujan and Expander Graphs Combinatorial 635 Case f(x) g(x) 1.1 1+bxm/2 + xm 1 bxm/2 + x3m − − 1.2 1+bxm/2 + xm b + xm/2 + x5m/2 1.3 1+bxm/2 + xm b bxm + x5m/2 − − 1.4 1+bxm/2 + xm b x3m/2 + x5m/2 − − − 1.5 1+bxm/2 + xm b + bx4m/2 + x5m/2 1.6 1+bxm/2 + xm 1+xm + x2m 1.7 1+bxm/2 + xm b + xm + x3m/2 − 1.8 1+bxm/2 + xm b bxm + x3m/2 − − − 1.9 a xm/3 + xm a xm/3 + x3m − − − 1.10 a xm/3 + xm 1+x2m/3 + x8m/3 − 1.11 a + xm/3 + xm a + ax2m/3 + x7m/3 1.12 a xm/3 + xm a ax4m/3 + x7m/3 − − 1.13 a xm/3 + xm a + x5m/3 + x7m/3 − − 1.14 a + xm/3 + xm 1+ax5m/3 + x2m 1.15 a xm/3 + xm a + ax4m/3 + x5m/3 − 1.16 1+bxm/4 + xm b + bx6m/4 + x11m/4 − − 1.17 1+bxm/4 + xm 1+bx9m/4 + x10m/4 Table 14.9.8 Table of polynomials such that g = fh with f and g monic trinomials over F3. 2.2 For tables of primitives of various kinds and weights. §3.4 For results on the weights of irreducible polynomials. §4.3 For results on the weights of primitive polynomials. §10.2 For discussion on bias and randomness of linear feedback shift register sequences. §14.1 For results of latin squares which are strongly related to orthogonal arrays. §14.5 For a discussion of block designs, which include orthogonal arrays. §15.1 Uses primitive polynomials to generate cyclic redundancy check and BCH codes. §15.4 For results on turbo codes. §17.3 For more discussion of applications of finite fields and polynomials. § References Cited: [146, 217, 356, 557, 558, 799, 824, 830, 831, 1360, 1455, 1489, 1587, 1588, 1619, 1724, 1789, 1790, 1831, 1942, 1992, 2072, 2081, 2167, 2202, 2352, 2437, 2511] 14.10 Ramanujan and expander graphs M. Ram Murty, Queen’s University Sebastian M. Cioab˘a, University of Delaware In the last two decades, the theory of Ramanujan graphs has gained prominence primarily for two reasons. First, from a practical viewpoint, they resolve an extremal problem in commu- nication network theory (see for example [268, 1532]). Second, from a more aesthetic view- point, they fuse diverse branches of pure mathematics, namely, number theory, representa- 636 Handbook of Finite Fields tion theory and algebraic geometry. The purpose of this survey is to unify some of the recent developments and expose certain open problems in the area. This survey is an expanded version of [2206] and is by no means an exhaustive one and demonstrates a highly number- theoretic bias. For other surveys, we refer the reader to [1532, 1920, 1965, 1966, 2523, 2837]. For a more up-to-date survey highlighting the connection between graph theory and auto- morphic representations, we refer the reader to Li’s recent survey article [1922]. 14.10.1 Graphs, adjacency matrices and eigenvalues 14.10.1 Definition A graph X is a pair (V,E) consisting of a vertex set V = V (X) and an edge set E = E(X) which is a multiset of unordered pairs of (not necessarily distinct) vertices. Each edge consists of two vertices that are called its endpoints.Aloop is an edge whose endpoints are equal. Multiple edges are edges having the same pair of endpoints. A simple graph is a graph having no loops nor multiple edges. If a graph has loops or multiple edges, we will call it a multigraph.Whentwoverticesu and v are endpoints of an edge, they are adjacent and write u v to indicate this fact. A directed graph Y is a pair (W, F ) consisting of a set of vertices∼ W and a multiset F of ordered pairs of vertices which are called arcs. 14.10.2 Remark All the graphs in this chapter are undirected unless stated explicitly otherwise. 14.10.3 Definition The degree of a vertex v of a graph X, denoted by deg(v), is the number of edges incident with v, where we count a loop with multiplicity 1. A graph X is k-regular if every vertex has degree k. 14.10.4 Proposition (Handshaking Lemma) For any simple graph X, deg(v)=2E(X) . v V (X) | | If X is a k-regular graph with n vertices, then E(X) = kn/2. ∈ | | 14.10.5 Definition An adjacency matrix A = A(X) of a graph X with n vertices is an n n matrix with rows and columns indexed by the vertices of X,wherethe(x, y)-th entry× equals the number of edges between vertex x and vertex y. As X is an undirected graph with n vertices, the matrix A(X) is symmetric and therefore, its eigenvalues λ1 λ2 λn are real. The multiset of eigenvalues of X is the spectrum of X. ≥ ≥···≥ 14.10.6 Remark We remark that the adjacency matrix defined above depends on the labeling of the vertices of X. Different labelings of the vertices of a graph X may possibly yield different adjacency matrices. However, all these adjacency matrices are similar to each other (by permutation matrices) and thus, their spectrum is the same. 14.10.7 Remark One can define an adjacency matrix of a directed graph Y =(W, A) similarly. Given a labeling of the vertices W of Y ,the(x, y)-th entry of the adjacency matrix corresponding to this labeling equals the number of arcs from x to y. Adjacency matrices of directed graphs may be non-symmetric. (1) (n 1) 14.10.8 Example The spectrum of the complete graph Kn on n vertices is (n 1) , ( 1) − , where the exponent signifies the multiplicity of the respective eigenvalue.− The− Petersen graph has spectrum 3(1), 1(5), 2(4). − 14.10.9 Theorem Let X be a graph on n vertices with maximum degree ∆and average degree d. Then d λ ∆and λ ∆for every 2 i n. ≤ 1 ≤ | i|≤ ≤ ≤ Combinatorial 637 14.10.10 Definition For a multigraph X,awalk of length r from x to y is a sequence x = v0,v1,...,vr = y of vertices of X such that vivi+1 is an edge of X for any i = 0, 1,...,r 1. The length of this walk is r.Aclosed walk is a walk where the start- ing vertex−x is the same as the last vertex y. 14.10.11 Definition A path is a walk with no repeated vertices. A cycle is a closed walk with no repeated vertices except the starting vertex. 14.10.12 Remark A word of caution must be inserted here. In graph theory literature, the distinction between a walk and a path is as we have defined it above. However, in number theory circles, the finer distinction is not made and one uses the word “path” to mean a “walk”; see for example, [2521, 2789]. 14.10.13 Definition A graph X is connected if for every two distinct vertices x and y,thereisa path from x to y. 14.10.14 Proposition For every graph X with adjacency matrix A and any integer r 1, the (x, y)-th entry of Ar equals the number of walks of length r between x and y. ≥ 14.10.15 Proposition The number of closed walks of length r in a graph X with n vertices equals λr + λr + + λr . 1 2 ··· n 14.10.16 Definition An independent set in a graph X is a subset of vertices that are pairwise non- adjacent. A graph X is bipartite if its vertex set can be partitioned into two independent sets A and B; X is complete bipartite and denoted by K A , B if it contains all the edges between A and B. | | | | 14.10.17 Proposition A graph is bipartite if and only if it does not contain any cycles of odd length. 14.10.18 Theorem If X is a k-regular and connected graph with n vertices, then λ1 = k and the multiplicity of k is 1 with the eigenspace of k spanned by the all 1 vector of dimension n. If X is a k-regular and connected graph, then X is bipartite if and only if λ = k. n − 14.10.19 Definition If X is a k-regular and connected graph, then the eigenvalue k of X is trivial. All other eigenvalues of X are non-trivial. Let λ(X) = max λi , where the maximum is taken over all non-trivial eigenvalues of X. The parameter λ(|X)isthe| second eigenvalue of X by some authors. The second largest eigenvalue of X is λ (X) and λ(X) λ (X). 2 ≥ 2 14.10.20 Definition The distance d(x, y)betweentwodistinctverticesx and y of a connected graph X is the length of a shortest path between x and y.Thediameter D of a connected graph X is the maximum of d(x, y), where the maximum is taken over all pairs of distinct vertices x = y of X. 14.10.21 Remark When k 3, if X is a k-regular and connected graph with n vertices and diameter D ≥ D 1 (k 1) 1 D,thenn 1+k + k(k 1) + + k(k 1) − =1+k −k 2 − and consequently, ≤ − ··· − · − (n 1)(k 2) log(n 1) log(k/(k 2)) D logk 1 − k − +1 > log(k−1) log(k −1) .
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