Abstract A perfect code C within a graph 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 is defined as a set of vertices where every vertex in 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.
Ahanjideh et al. (2026) studied this question.
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