On bipartite distance-regular Cayley graphs with small diameter

TL;DR

This paper investigates bipartite distance-regular Cayley graphs with small diameters, providing conditions for their construction, extending known results, and exploring specific cases involving difference sets and group structures.

Contribution

It offers new sufficient conditions for constructing bipartite Cayley graphs with small diameters and extends existing results to diameter four.

Findings

Conditions for bipartite Cayley graph construction on semidirect products.

Extension of difference set results to diameter four.

Analysis of specific groups like dihedral groups.

Abstract

We study bipartite distance-regular Cayley graphs with diameter three or four. We give sufficient conditions under which a bipartite Cayley graph can be constructed on the semidirect product of a group -- the part of this bipartite Cayley graph which contains the identity element -- and $\mathbb{Z}_{2}$. We apply this to the case of bipartite distance-regular Cayley graphs with diameter three, and consider cases where the sufficient conditions are not satisfied for some specific groups such as the dihedral group. We also extend a result by Miklavi\v{c} and Poto\v{c}nik that relates difference sets to bipartite distance-regular Cayley graphs with diameter three to the case of diameter four. This new case involves certain partial geometric difference sets and -- in the antipodal case -- relative difference sets.