Diameter of Ramanujan Graphs and Random Cayley Graphs

We study the diameter of LPS Ramanujan graphs $X_{p,q}$. 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 $X_{p,q}$. We also construct an infinite family of $(p+1)$-regular LPS Ramanujan graphs $X_{p,m}$ such that the diameter of these graphs is greater than or equal to $\lfloor (4/3)\log_{p}(n) \rfloor$. 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 $(1+\epsilon)\log_{k-1}(n)$. 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.