Graduate Student Number Theory Seminar

This talk is based primarily on the paper Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees by Marcus, Spielman, and Srivastava [6]. By proving a variant of a conjecture of Bilu and Linial [1], they show that there exist infinite families of d-regular bipartite Ramanujan graphs for d ≥ 3. Of particular interest in the paper by Marcus, Spielman, and Srivastava is their “method of interlacing polynomials”[6]. 1. Basic Definitions Think of the adjacency matrix of a graph G as an adjacency operator. Then an eigenvalue of the matrix is an eigenvalue of the operator. If G is d-regular, then the largest eigenvalue is d. If G is connected, then the next eigenvalue is smaller than d. Definition 1. For a d-regular graph G with eigenvalues d = λ1 ≥ λ2 ≥ . . . λn, if |λ2|, . . . , |λn| ≤ 2 √ d− 1, then G is said to be Ramanujan. If G is bipartite, then the spectrum is symmetric about the origin; therefore if G is a d-regular bipartite graph, then the smallest eigenvalue is −d. Definition 2. For a d-regular bipartite graph G with eigenvalues d = λ1 ≥ λ2 ≥ . . . λn = −d, if |λ2|, . . . , |λn−1| ≤ 2 √ d− 1, then G is said to be bipartite Ramanujan. A simple example of a d-regular Ramanujan graph is Kd+1, the complete graph. A simple example of a d-regular bipartite Ramanujan graph is Kd,d, the complete bipartite graph. The value 2 √ d− 1 arises naturally. Theorem 1 (Alon-Boppana). For all fixed ε > 0, there exists N > 0 s.t. for all graphs on n > N vertices, we have λ2 > 2 √ d− 1− ε. Date: October 17, 2013. 2 Definition 3. If G is a graph on n vertices with maximum degree d and for every W ⊂ V with |W | ≤ n 2 the inequality |N(W )| ≥ λ|W | holds (where N(W ) is the neighborhood of W in V \W ), then G is a (n, d, λ)-expander. Simply speaking, expander graphs are sparse yet highly connected d-regular graphs. Because of these nice properties, expander graphs have many applications in engineering and computer science from network design to cryptography. A large spectral gap indicates high connectivity. Ramanujan graphs are “good quality” expanders. Theorem 2. For the following values of d, there exist infinite families of d-regular Ramanujan graphs: (1) d = p+ 1, p an odd prime (1988 Lubotzky-Phillips-Sarnak [5] and 1988 Margulis [8]) (2) d = 2 + 1 = 3 (1992 Chiu [2]), (3) d = q + 1, q a prime power(1994 Morgenstern [9]). Theorem 3 (Marcus, Spielman, and Srivastava[6]). For all d ≥ 3, there are infinitely many d-regular, bipartite Ramanujan graphs. While the construction of Ramanujan graphs is fairly simple, proving they have the desired properties is not and relies heavily on group theory, modular forms, and even the Riemann Hypothesis for curves over finite fields. The name “Ramanujan” comes from the constructions’ dependence on Ramanujan’s conjecture (solved by Eichler in 1954) concerning coefficients of modular forms with weight 2. Eichler related the eigenvalues of Hecke operators Tm acting on spaces of cusp forms to the zeros of zeta functions of modular curves over the fields Fp. For m prime, varying the space on which the Hecke operators Tm act, we obtain a large family of Ramanujan graphs. For Tm, m not prime, we are able to associate an “almost Ramanujan” graph.[10]. Curiously some applications actually require bipartite or non-bipartite Ramanujan graphs. Explicit constructions of the error correcting codes of Sipser and Spielman actually require non-bipartite expanders [11] whereas improvements of this construction require bipartite Ramanujan expanders [12]. Graduate Student Number Theory Seminar Delcourt 3 2. 2-Lifting In 2006 Bilu and Linial suggested constructing Ramanujan graphs by iteratively applying 2-lifts to a base graph[1]. Given a graph G = (V,E), we construct a 2-lift of G, say G̃ = (Ṽ , Ẽ), in the following manner. We can think of G̃ as being a covering space of G. Let vertex set of G̃ be a multiset containing two copies of V , say Ṽ = V0 ∪ V1. Each pair of vertices in Ṽ is the fiber over the original vertex in V , and each edge in E corresponds to two edges in Ẽ. For each edge uv ∈ E, if {u0, u1} is the fiber of u and {v0, v1} is the fiber of v, then G̃ contains either the pair of edges u0v0 and u1v1 or the pair of edges u0v1 and u1v0. Note that if only edges of the first type appear, then G̃ consists of two copies of G. If only edges of the second type appear, then G̃ is the double cover of G. Also, a 2-lift of a bipartite graph by definition is also bipartite. Bilu and Linial also introduce the notion of signings of E to calculate the eigenvalues of G̃. Let s : E → {±1} with s(uv) = 1 if the fiber of uv is a pair of the first type and s(uv) = −1 if the fiber of uv is a pair of the second type. Definition 4. Let G be a graph with adjacency matrix A. The entries of the signed adjacency matrix As associated with a 2-lift G̃ are (As)uv = s(uv), if uv ∈ E 0, otherwise. The eigenvalues of G̃ are related to A and As. We think of the eigenvalues of A as “old” eigenvalues and the eigenvalues of As as “new” eigenvalues of G̃. Lemma 1. Let A be the adjacency matrix of a graph G on n vertices, and As the signed adjacency matrix associated with a 2-lift G̃. Then every eigenvalue of A and every eigenvalue of As are eigenvalues of G̃. Furthermore, the multiplicity of each eigenvalue of G̃ is the sum of the multiplicities in A and As. Proof. The adjacency matrix of G̃ is à = A1 A2 A2 A1  Graduate Student Number Theory Seminar Delcourt 4 whereA1 is the adjacency matrix of (V, s −1(1)) andA2 is the adjacency matrix of (V, s −1(−1)). Note that A = A1 + A2 and As = A1 − A2. If v is an eigenvector of A with eigenvalue μ, then v̂ = (v v) is an eigenvector of à with eigenvalue μ. If u is an eigenvector of As with eigenvalue λ, then û = (u−u) is an eigenvector of à with eigenvalue λ. Because the û’s and v̂’s are perpendicular and there are 2n of them, they span all eigenvectors of Ã. Example 1. Consider the weighted graph and the corresponding 2-lift in Figure 1 below. Then A =  0 1 1 1 0 1 1 1 0  , As =  0 1 1 1 0 −1 1 −1 0  , A1 =  0 1 1 1 0 0 1 0 0  , and A2 =  0 0 0 0 0 1 0 1 0  .