Regular sets in Cayley sum graphs on generalized dicyclic groups

TL;DR

This paper investigates the structure of regular sets in Cayley sum graphs on generalized dicyclic groups, characterizing possible regularity parameters for subgroups as regular sets with specific connection sets.

Contribution

It provides a classification of all possible regularity parameters for subgroups to be regular sets in Cayley sum graphs on generalized dicyclic groups.

Findings

Determines all possible (,) parameters for subgroup regular sets.

Characterizes connection sets that realize each regularity pattern.

Extends understanding of regular sets in Cayley sum graphs on complex groups.

Abstract

For a graph $\Gamma=(V(\Gamma),E(\Gamma))$, a subset $C$ of $V(\Gamma)$ is called an $(\alpha,\beta)$-regular set in $\Gamma$, if every vertex of $C$ is adjacent to exactly $\alpha$ vertices of $C$ and every vertex of $V(\Gamma)\setminus C$ is adjacent to exactly $\beta$ vertices of $C$. In particular, if $C$ is an $(\alpha,\beta)$-regular set in some Cayley sum graph of a finite group $G$ with connection set $S$, then $C$ is called an $(\alpha,\beta)$-regular set of $G$. In this paper, we consider a generalized dicyclic group $G$ and for each subgroup $H$ of $G$, by giving an appropriate connection set $S$, we determine each possibility for $(\alpha,\beta)$ such that $H$ is an $(\alpha,\beta)$-regular set of $G$.