On directed strongly regular Cayley graphs over non-abelian groups with an abelian subgroup of index $2$

TL;DR

This paper characterizes directed strongly regular Cayley graphs over non-abelian groups with an abelian subgroup of index 2, extending previous results and providing necessary conditions for their existence.

Contribution

It provides a characterization and necessary conditions for directed strongly regular Cayley graphs over a broader class of non-abelian groups.

Findings

Necessary conditions for such Cayley graphs to be directed strongly regular.

Complete characterization under certain conditions.

Extension of previous results by He and Zhang (2019).

Abstract

In 1988, Duval introduced the concept of directed strongly regular graphs, which can be viewed as a directed graph version of strongly regular graphs. Such directed graphs have similar structural and algebraic properties to strongly regular graphs. In the past three decades, it was found that Cayley graphs, especially those over dihedral groups, play a key role in the construction of directed strongly regular graphs. In this paper, we focus on the characterization of directed strongly regular Cayley graphs over more general groups. Let $G$ be a non-abelian group with an abelian subgroup of index $2$. We give some necessary conditions for a Cayley graph over $G$ to be directed strongly regular, and characterize the directed strongly regular Cayley graphs over $G$ satisfying specified conditions. This extends some previous results of He and Zhang (2019).