Regular graphs of large girth and arbitrary degree

TL;DR

This paper constructs infinite families of large girth, regular graphs for any degree greater than 9, using Cayley graphs related to PGL_2(q) and arithmetic of quaternions, improving previous bounds for non-prime degrees.

Contribution

It introduces new constructions of large girth regular graphs for arbitrary degrees > 9, extending and improving prior results for non-prime and power-of-two degrees.

Findings

Constructed infinite families of d+1-regular graphs with girth > log_d |G_n|

Graphs are Cayley graphs on PGL_2(q) related to quaternion arithmetic

Improves previous girth bounds for non-prime and power-of-two degrees

Abstract

For every integer d > 9, we construct infinite families {G_n}_n of d+1-regular graphs which have a large girth > log_d |G_n|, and for d large enough > 1,33 log_d |G_n|. These are Cayley graphs on PGL_2(q) for a special set of d+1 generators whose choice is related to the arithmetic of integral quaternions. These graphs are inspired by the Ramanujan graphs of Lubotzky-Philips-Sarnak and Margulis, with which they coincide when d is prime. When d is not equal to the power of an odd prime, this improves the previous construction of Imrich in 1984 where he obtained infinite families {I_n}_n of d+1-regular graphs, realized as Cayley graphs on SL_2(q), and which are displaying a girth > 0,48 log_d |I_n|. And when d is equal to a power of 2, this improves a construction by Morgenstern in 1994 where certain families {M_n}_n of 2^k+1-regular graphs were shown to have a girth > 2/3 log_d |M_n|.