Regular set in Cayley sum mgraph

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≤ α ≤ |NSq(G)∩ H| and 0≤ β ≤ L(H), where L(H)=|H|, if Sq(G) ⊆ H and L(H)=|NSq(G)∩ H|, otherwise. As a consequence of our result we give a very brief proof for the main results in {mama, ma}. Also, we consider the dihedral group G=D₂ₙ 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.