Regular set in Cayley sum mgraph

Abstract

A subset $C$ of the vertex set of a graph $Γ$ is said to be $(α,β)$-regular if $C$ induces an $α$-regular subgraph and every vertex outside $C$ is adjacent to exactly $β$ vertices in $C$. In particular, if $C$ is an $(α,β)$-regular set in some Cayley sum graph of a finite group $G$ with connection set $S$, then $C$ is called an $(α,β)$-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 $(α,β)$-regular set of $G$, for each $0\leq α\leq |$NSq$(G)\cap H|$ and $0\leq β\leq \mathcal{L}(H)$, 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 $(α, β)$, where $H$ is an $(α,β)$-regular set of $G$.