Abelian Girth and Girth

TL;DR

This paper establishes a lower bound relationship between abelian girth and girth in graphs, and derives an analogue of the Moore bound for abelian girth in regular graphs, suggesting potential improvements for graph bounds.

Contribution

It introduces a new lower bound for abelian girth relative to girth and formulates an analogue of the Moore bound for abelian girth in regular graphs.

Findings

Abelian girth is at least three times the girth.

An analogue of the Moore bound for abelian girth is proven.

Potential for improving graph bounds by focusing on abelian girth.

Abstract

We show that the abelian girth of a graph is at least three times its girth. We prove an analogue of the Moore bound for the abelian girth of regular graphs, where the degree of the graph is fixed and the number of vertices is large. We conclude that one could try to improve the Moore bound for graphs of fixed degree and many vertices by trying to improve its analogue concerning the abelian girth.