On perfect codes and total perfect codes in Cayley sum graph

A perfect code C within a graph Γ Γ consists of an independent set of vertices such that every vertex not in C is connected to exactly one vertex within C . A total perfect code C in Γ Γ is defined as a set of vertices where every vertex in Γ Γ is adjacent to a unique vertex in C . Let G represent a finite group, and let X be a normal subset of G . The Cayley sum graph CS ( G , X ) is constructed with vertex set G , where two vertices g and h are adjacent if gh is included in X and g ≠ h g ≠ h . In this paper, we demonstrate that a group G is abelian if and only if every subgroup of G qualifies as either a perfect code or a total perfect code in some Cayley sum graph associated with G .