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. 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 \(0\le \alpha \le |\) NSq \((G)\cap H|\) and \(0\le \beta \le {\mathcal {L}}(H)\) , where \({\mathcal {L}}(H)=|H|\) , if Sq \((G) \subseteq H\) and \({\mathcal {L}}(H)=|\) NSq \((G)\cap H|\) , otherwise. As a consequence we easily get that H is a (0, 1)-regular, if and only if either Sq \((G)\subseteq H\) or NSq \((G)\cap H\not =\emptyset \) . The proof of this result is given by X. Ma, K. Wang, and Y. Yang in 2022, in a longer method. Also, X. Ma, M. Feng, and K. Wang in 2020, gave a sufficient and necessary condition for a subgroup H to be a (0, 1)-regular subgroup of an abelian group G. Our new result makes that result much easier to gain. 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.
