Regular sets in Cayley sum graphs on generalized dicyclic groups

For a graph $\mathrm{\Gamma }=(V(\mathrm{\Gamma }),E(\mathrm{\Gamma }))$, a subset $C$ of $V(\mathrm{\Gamma })$ is called an $(\alpha ,\beta )$-regular set in $\mathrm{\Gamma }$, if every vertex of $C$ is adjacent to exactly $\alpha$ vertices of $C$ and every vertex of $V(\mathrm{\Gamma })\setminus C$ 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 considered a generalized dicyclic group $G$ and for each subgroup $H$ of $G$, by giving an appropriate connection set $S$, we determined each possibility for $(\alpha ,\beta )$ such that $H$ is an $(\alpha ,\beta )$-regular set of $G$.