A formulation of a (q+1,8)-cage

TL;DR

This paper introduces a new formulation for constructing (q+1,8)-cages, enabling the creation of smaller graphs with specified degree and girth, advancing graph theory constructions.

Contribution

The paper presents a novel formulation for (q+1,8)-cages that facilitates the construction of smaller (k,g)-graphs for specific parameters, expanding existing graph construction methods.

Findings

Constructed smaller (k,g)-graphs for k=q-1, q and g=7,8.

Developed smaller (q,8)-graphs for even prime powers q.

Provided a new formulation for known (q+1,8)-cages.

Abstract

Let $q\ge 2$ be a prime power. In this note we present a formulation for obtaining the known $(q+1,8)$-cages which has allowed us to construct small $(k,g)$--graphs for $k=q-1, q$ and $g=7,8$. Furthermore, we also obtain smaller $(q,8)$-graphs for even prime power $q$.