A construction of small regular bipartite graphs of girth 8

Camino Balbuena

doi:10.46298/dmtcs.461

Abstract

Let q be a prime a power and k an integer such that 3 ≤ k ≤ q. In this paper we present a method using Latin squares to construct adjacency matrices of k-regular bipartite graphs of girth 8 on 2(kq2 -- q) vertices. Some of these graphs have the smallest number of vertices among the known regular graphs with girth 8.

Highlights

Throughout this paper, only undirected simple graphs without loops or multiple edges are considered.Unless otherwise stated, we follow the books by Godsil and Royle [16] and by Lint and Wilson [21] for terminology and definitions
A cage is a k-regular graph with girth g having the smallest possible number of vertices
A Latin square has clearly girth g = ∞ because the position matrices of its elements are permutation matrices yielding the incidence matrix of a partial plane consisting in a set of parallel lines

Summary

Introduction

Throughout this paper, only undirected simple graphs without loops or multiple edges are considered. As already mentioned in the Introduction, our main aim is to obtain incidence matrices of bipartite k-regular graphs of girth 8 with small excess Such incidence matrices may be seen as incidence matrices of partial planes which will be obtained by identifying row i of Pz(Aα) as line i(α), and column j of Pz(Aα) as point j(z), for any matrix Aα ∈ F and z ∈ S − 0. Ar} of the same number of columns whose elements are subsets of a set of symbols S is said to have girth g if the position matrix of F is the incidence matrix of a bipartite graph of girth g. A Latin square has clearly girth g = ∞ because the position matrices of its elements are permutation matrices yielding the incidence matrix of a partial plane consisting in a set of parallel lines (since they have no common point). Our immediate goal is to derive a method for constructing a family of matrices with girth 8, because the position matrix of this family will be the incidence matrix of a bipartite graph of girth 8

Method

Conclusion