Regular sets in Cayley sum graphs on generalized dicyclic groups

Abstract

For a graph $\hat{I}“=(V(\hat{I}“),E(\hat{I}“))$, a subset $C$ of $V(\hat{I}“)$ is called an $(\hat{I}\pm ,{\hat{I}}^2)$-regular set in $\hat{I}“$, if every vertex of $C$ is adjacent to exactly $\hat{I}\pm$ vertices of $C$ and every vertex of $V(\hat{I}“)\setminus C$ is adjacent to exactly ${\hat{I}}^2$ vertices of $C$. In particular, if $C$ is an $(\hat{I}\pm ,{\hat{I}}^2)$-regular set in some Cayley sum graph of a finite group $G$ with connection set $S$, then $C$ is called an $(\hat{I}\pm ,{\hat{I}}^2)$-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 $(\hat{I}\pm ,{\hat{I}}^2)$ such that $H$ is an $(\hat{I}\pm ,{\hat{I}}^2)$-regular set of $G$.