Distance-regular Cayley graphs over $\mathbb{Z}_{p^s}\oplus\mathbb{Z}_{p}$

TL;DR

This paper classifies all distance-regular Cayley graphs over the group rac{rac{p^s}{p}} with p an odd prime, showing they are either complete, multipartite, or line graphs of certain designs.

Contribution

It provides a complete classification of distance-regular Cayley graphs over rac{rac{p^s}{p}} for odd primes, extending previous characterizations.

Findings

Graphs are isomorphic to complete graphs, multipartite graphs, or line graphs of transversal designs.

All such graphs are explicitly characterized.

The classification covers all cases for odd prime p.

Abstract

In [Distrance-regular Cayley graphs on dihedral groups, J. Combin. Theory Ser B 97 (2007) 14--33], Miklavi\v{c} and Poto\v{c}nik proposed the problem of characterizing distance-regular Cayley graphs, which can be viewed as an extension of the problem of identifying strongly regular Cayley graphs, or equivalently, regular partial difference sets. In this paper, all distance-regular Cayley graphs over $\mathbb{Z}_{p^s}\oplus\mathbb{Z}_{p}$ with $p$ being an odd prime are determined. It is shown that every such graph is isomorphic to a complete graph, a complete multipartite graph, or the line graph of a transversal design $TD(r,p)$ with $2\leq r\leq p-1$.