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)logp(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)logp(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+ϵ) logk–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)logk–1(n) and logk–1(n), respectively.
Mathematics Subject Classification (2010)
05C25 20–40 05C35
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