Regular set in Cayley sum mgraph

A subset C of the vertex set of a graph is said to be (, ) -regular if C induces an -regular subgraph and every vertex outside C is adjacent to exactly vertices in C. In particular, if C is an (, ) -regular set in some Cayley sum graph of a finite group G with connection set S, then C is called an (, ) -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 (, ) -regular set of G, for each 0 |NSq (G) H| and 0 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=D₂₍ and for each subgroup H of G, by giving an appropriate connection set S, we determine each possibility for (, ), where H is an (, ) -regular set of G.