Families of small regular graphs of girth 7

TL;DR

This paper introduces combinatorial methods to construct new infinite families of small regular graphs with girth 7, expanding the known classes of cages derived from generalized polygons.

Contribution

It presents novel combinatorial operations to generate infinite families of small regular graphs of girth 7 from existing cages based on generalized quadrangles.

Findings

Constructed new infinite families of girth 7 graphs

Extended the methodology for building cages from generalized polygons

Provided explicit combinatorial operations for graph construction

Abstract

The first known families of cages arised from the incidence graphs of generalized polygons of order $q$, $q$ a prime power. In particular, $(q+1,6)$--cages have been obtained from the projective planes of order $q$. Morever, infinite families of small regular graphs of girth 5 have been constructed performing algebraic operations on $\mathbb{F}_q$. In this paper, we introduce some combinatorial operations to construct new infinite families of small regular graphs of girth 7 from the $(q+1,8)$--cages arising from the generalized quadrangles of order $q$, $q$ a prime power.