Diameter of Ramanujan Graphs and Random Cayley Graphs

Abstract

We study the diameter of LPS Ramanujan graphs Xp,q. We show that the diameter of the bipartite Ramanujan graphs is greater than (4/3) log p (n) + O(1) where n is the number of vertices of Xp,q. We also construct an infinite family of (p + 1)-regular LPS Ramanujan graphs Xp,m such that the diameter of these graphs is greater than or equal to (4/3) log p (n) . On the other hand, for any k-regular Ramanujan graph we show that the distance of only a tiny fraction of all pairs of vertices is greater than (1 + ) log k−1 (n). We also have some numerical experiments for LPS Ramanujan graphs and random Cayley graphs which suggest that the diameters are asymptotically (4/3) log k−1 (n) and log k−1 (n), respectively.

Quantum Chaos on Random Cayley Graphs of SL 2[Z/pZ]

“…This generalizes the celebrated result of Bourgain and Gamburd on the expansion of random Cayley graphs on SL 2 [Z/pZ][BG08]. As a consequence of the work of Lubetzky and Peres[LP15] or the second author[Tal15a], our conjecture implies that a random symmetric set of generators with 2d elements represents almost all elements of SL…”

“…On the other hand, the diameter of the Cayley graph with LPS generators that is a special fixed generator family of Cayley graphs is not optimal. The second named author in [Tal15a], showed that the diameter of a family of LPS Ramanujan graphs is greater than 4/3 log d−1 (n) where n is the number of vertices and d is the degree of the graph. This lower bound is related to the repulsion of integral points lying on quadrics.…”

Cutoff on all Ramanujan graphs

“…Sardari in an insightful paper [31] posted to the arXiv a few months after the initial posting of the present paper. For a certain infinite family of (p + 1)-regular n-vertex Ramanujan graphs, Sardari [31] shows that the diameter is at least 4 3 log p (n) and also gives an alternative proof of the first part of Corollary 2.…”

An Explicit Infinite Family of $$\mathbb{M}$$-Vertex Graphs with Maximum Degree K and Diameter $$\left[ {1 + o\left( 1 \right)} \right]{\log _{K - 1}\mathbb{M}}$$ for Each K − 1 a Prime Power

“…Then Theorem 1.6 gives us the following: Theorem 1.6 (Theorem 1.3 in [14]). First, let k be an integer at least 3, and let G be a k -regular nonbipartite Ramanujan graph.…”

“…Another critical property of Ramanujan graphs that we will use is Theorem 1.6 below, which was very recently established by N. Sardari [14], and says that the average distance between pairs of vertices in a Ramanujan graph is small. More precisely: Definition 1.5.…”