On distance-regular Cayley graphs on abelian groups

TL;DR

This paper classifies all distance-regular Cayley graphs on finite abelian groups with specific generating sets, expanding understanding of their structure and properties.

Contribution

It provides a complete classification of distance-regular Cayley graphs on abelian groups under particular generating set conditions.

Findings

Complete classification of such graphs achieved

Identification of structural properties of the generating sets

Clarification of conditions for distance-regularity in Cayley graphs

Abstract

Let $G$ denote a finite abelian group with identity 1 and let $S$ denote an inverse-closed subset of $G \setminus {1}$, which generates $G$ and for which there exists $s \in S$, such that $\la S \setminus \{s,s^{-1}\} \ra \ne G$. In this paper we obtain the complete classification of distance-regular Cayley graphs $\cay(G;S)$ for such pairs of $G$ and $S$.