Small Regular Graphs of Girth 7

M Abreu,C Balbuena,G Araujo-Pardo,J Salas,D Labbate

doi:10.37236/4205

Abstract

In this paper, we construct new infinite families of regular graphs of girth 7 of smallest order known so far. Our constructions are based on combinatorial and geometric properties of $(q+1,8)$-cages, for $q$ a prime power. We remove vertices from such cages and add matchings among the vertices of minimum degree to achieve regularity in the new graphs. We obtain $(q+1)$-regular graphs of girth 7 and order $2q^3+q^2+2q$ for each even prime power $q \ge 4$, and of order $2q^3+2q^2-q+1$ for each odd prime power $q\ge 5$. A corrigendum was added to this paper on 21 June 2016.

Highlights

Throughout this paper, only undirected simple graphs without loops or multiple edges are considered
Let G be a graph with vertex set V = V (G) and edge set E = E(G)
We denote the subgraph of G induced by a subset U ⊂ V (G) as G[U ], and it is the graph with V (G[U ]) = U and for any u, v ∈ V (G[U ]) the edge uv belongs to E(G[U ]) if and only if uv ∈ E(G)

Summary

Introduction

Throughout this paper, only undirected simple graphs without loops or multiple edges are considered. We construct a family of (q + 1, 7)-graphs of order 2q3 + q2 + 2q obtained from a (q + 1, 8)-cage Γq for each even prime power q 4.

Results

Conclusion