Distance-regular Cayley graphs over dicyclic groups

TL;DR

This paper classifies all distance-regular Cayley graphs over dicyclic groups, showing they are either complete, multipartite, or specific bipartite graphs with diameter three, advancing understanding of their structure.

Contribution

It provides a complete classification of distance-regular Cayley graphs on dicyclic groups, identifying their possible structures and conditions.

Findings

Distance-regular Cayley graphs on dicyclic groups are complete, multipartite, or bipartite with diameter 3.

Characterization of these graphs helps in understanding their algebraic and combinatorial properties.

The results extend the classification of Cayley graphs with regularity conditions.

Abstract

The characterization of distance-regular Cayley graphs originated from the problem of identifying strongly regular Cayley graphs, or equivalently, regular partial difference sets. In this paper, a classification of distance-regular Cayley graphs on dicyclic groups is obtained. More specifically, it is shown that every distance-regular Cayley graph on a dicyclic group is a complete graph, a complete multipartite graph, or a non-antipodal bipartite distance-regular graph with diameter $3$ satisfying some additional conditions.