On Highly-regular graphs

TL;DR

This paper explores highly-regular graphs, a generalization of distance-regular graphs, providing characterizations, constructions, and property generalizations to deepen understanding of their combinatorial structure.

Contribution

It introduces new characterizations, constructions, and generalizations for highly-regular graphs, expanding the theoretical framework beyond distance-regular graphs.

Findings

Characterization of distance-regular graphs via highly-regular graph parameters

Two new constructions of highly-regular graphs

Generalization of intersection number properties

Abstract

Highly-regular graphs can be regarded as a combinatorial generalization of distance-regular graphs. From this standpoint, we study combinatorial aspects of highly-regular graphs. As a result, we give the following three main results in this paper. Firstly, we give a characterization of a distance-regular graph by using the index and diameter of a highly-regular graph. Secondly, we give two constructions of highly-regular graphs. Finally, we generalize well-known properties of the intersection numbers of a distance-regular graph.