Regular graphs of large girth and arbitrary degree

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For every integer d≥10, we construct infinite families {Gn}n∈ℕ of d+1-regular graphs which have a large girth ≥logd|Gn|, and for d large enough ≥1.33 · logd|Gn|. These are Cayley graphs on PGL2(Fq) 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 a 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 {In}n∈ℕ of d + 1-regular graphs, realized as Cayley graphs on SL2(Fq), and which are displaying a girth ≥0.48·logd|In|. And when d is equal to a power of 2, this improves a construction by Morgenstern in 1994 where certain families {Mn}n∈N of 2k +1-regular graphs were shown to have girth ≥2/3·log2k|Mn|.

Original languageEnglish
Pages (from-to)407-426
Number of pages20
Issue number4
Publication statusPublished - 2014 Aug 1


  • 05C25
  • 05C38


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