Diameter of Ramanujan Graphs and Random Cayley Graphs

We study the diameter of LPS Ramanujan graphs X-p,X-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 X-p,X-q. We also construct an infinite family of (p+1)-regular LPS Ramanujan graphs X-p,X-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 only a tiny fraction of all pairs of vertices have distance greater than (1+E) 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.