Diameter of Ramanujan Graphs and Random Cayley Graphs

TL;DR

This paper investigates the diameter of Ramanujan graphs, establishing lower bounds and asymptotic behaviors, and compares these with random Cayley graphs through theoretical results and numerical experiments.

Contribution

It provides new lower bounds on the diameter of LPS Ramanujan graphs and compares their diameter growth with that of random Cayley graphs.

Findings

Diameter of bipartite Ramanujan graphs exceeds (4/3) log_p(n)

Constructed infinite family of Ramanujan graphs with diameter ≥ (4/3) log_p(n)

Numerical experiments suggest diameters grow as (4/3) log_{k-1}(n) for Ramanujan graphs

Abstract

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.