Girth-regular graphs

TL;DR

This paper introduces girth-regular graphs, a generalization of symmetry concepts in graph theory, and characterizes cubic girth-regular graphs with small girth, expanding understanding of their structural properties.

Contribution

It defines girth-regular graphs, explores extremal signature cases, and characterizes cubic girth-regular graphs for girth up to 5, advancing the theory of symmetric graph structures.

Findings

Girth-regularity generalizes vertex transitivity and distance-regularity.

Results on extremal signatures for general girth-regular graphs.

Complete characterization of cubic girth-regular graphs with girth ≤ 5.

Abstract

We introduce a notion of a girth-regular graph as a $k$-regular graph for which there exists a non-descending sequence $(a_1, a_2, \dots, a_k)$ (called the signature) giving, for every vertex $u$ of the graph, the number of girth cycles the edges with end-vertex $u$ lie on. Girth-regularity generalises two very different aspects of symmetry in graph theory: that of vertex transitivity and that of distance-regularity. For general girth-regular graphs, we give some results on the extremal cases of signatures. We then focus on the cubic case and provide a characterisation of cubic girth-regular graphs of girth up to $5$.