## Abstract

For every integer d≥10, we construct infinite families {G_{n}}_{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}(F_{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 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 {I_{n}}_{n∈ℕ} of d + 1-regular graphs, realized as Cayley graphs on SL_{2}(F_{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∈N} of 2^{k} +1-regular graphs were shown to have girth ≥2/3·log_{2}^{k}|M_{n}|.

Original language | English |
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Pages (from-to) | 407-426 |

Number of pages | 20 |

Journal | Combinatorica |

Volume | 34 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2014 Aug 1 |

Externally published | Yes |

## Keywords

- 05C25
- 05C38

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Mathematics