Regular set in Cayley sum mgraph

TL;DR

This paper investigates regular sets in Cayley sum graphs of finite groups, characterizing when subgroups are regular sets and exploring their properties in abelian and dihedral groups.

Contribution

It provides a characterization of subgroup regular sets in finite abelian groups and analyzes regular sets in dihedral groups with explicit connection sets.

Findings

Subgroups of finite abelian groups can be regular sets under specific conditions.

The paper offers a brief proof of key results from prior studies.

It determines all possible regular set parameters in dihedral groups.

Abstract

A subset $C$ of the vertex set of a graph $\Gamma$ is said to be $(\alpha,\beta)$-regular if $C$ induces an $\alpha$-regular subgraph and every vertex outside $C$ is adjacent to exactly $\beta$ vertices in $C$. 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 $G$ and a $(0,1)$-regular set is called a perfect code of $G$. By Sq$(G)$ and NSq$(G)$ we mean the set of all square elements and non-square elements of $G$. As one of the main results in this note, we show that a subgroup $H$ of a finite abelian group $G$ is an $(\alpha,\beta)$-regular set of $G$, for each $0\leq \alpha \leq |$NSq$(G)\cap H|$ and $0\leq \beta \leq \mathcal{L}(H)$, where $\mathcal{L}(H)=|H|$, if Sq$(G) \subseteq H$ and $\mathcal{L}(H)=|$NSq$(G)\cap H|$, otherwise. As a consequence of our result we give a very brief proof for the main results in \cite{mama, ma}. Also, we consider the dihedral group $G=D_{2n} $ and for each subgroup $H $ of $G$, by giving an appropriate connection set $S$, we determine each possibility for $(\alpha, \beta)$, where $H$ is an $(\alpha,\beta)$-regular set of $G$.