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Diameter of Ramanujan Graphs and Random Cayley Graphs

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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)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.

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  1. Van Vleck Visiting Assistant Professor, University of Wisconsin, Madison, USA

    Naser T. Sardari

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  1. Naser T. Sardari
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Correspondence to Naser T. Sardari.

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Sardari, N.T. Diameter of Ramanujan Graphs and Random Cayley Graphs. Combinatorica 39, 427–446 (2019). https://doi.org/10.1007/s00493-017-3605-0

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  • DOI: https://doi.org/10.1007/s00493-017-3605-0

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