An explicit formula for obtaining $(q+1,8)$-cages and others small regular graphs of girth 8

TL;DR

This paper provides an explicit formula for constructing $(q+1,8)$-cages and derives small regular graphs of girth 8 by removing specific dominating sets, improving known minimal vertex counts.

Contribution

It introduces a new explicit formula for $(q+1,8)$-cages and constructs smaller regular graphs of girth 8 by modifying these cages.

Findings

Explicit formula for $(q+1,8)$-cages using graphical terminology.

Construction of smaller regular graphs of girth 8 with minimal vertices.

New methods to derive graphs with specific regularity and girth properties.

Abstract

Let $q$ be a prime power; $(q+1,8)$-cages have been constructed as incidence graphs of a non-degenerate quadric surface in projective 4-space $P(4, q)$. The first contribution of this paper is a construction of these graphs in an alternative way by means of an explicit formula using graphical terminology. Furthermore by removing some specific perfect dominating sets from a $(q+1,8)$-cage we derive $k$-regular graphs of girth 8 for $k= q-1$ and $k=q$, having the smallest number of vertices known so far.