Let G be a finite abelian group with identity 0. For an integer n ! 2, the additive power graph C apq ðGÞ of G is the simple undirected graph with vertex set G in which two distinct vertices x and y are adjacent if and only if x þ y ¼ nt for some t 2 G with nt 6 ¼ 0: When n ¼ 2, the additive power graph has been studied in the name of square graph of finite abelian groups. In this paper, we study the additive power graph of G with n ¼ 3 and name the graph as the cubic power graph. The cubic power graph of G is denoted by C cpg ðGÞ: More specifically, we obtain the diameter and the girth of the graph C cpg ðGÞ and its complement C cpg ðGÞ: Using these, we obtain a condition for C cpg ðGÞ and its complement C cpg ðGÞ to be self-centered. Also, we obtain the independence number, the clique number and the chromatic number of C cpg ðGÞ and its complement C cpg ðGÞ and hence we prove that C cpg ðGÞ and its complement C cpg ðGÞ are weakly perfect. Also, we discuss about the perfectness of C cpg ðGÞ: At last, we obtain a condition for C cpg ðGÞ and its complement C cpg ðGÞ to be vertex pancyclic.

Let G be a finite abelian group, written additively, and H a subgroup of G. The subgroup sum graph Γ G,H is the graph with vertex set G, in which two distinct vertices x and y are joined if x + y ∈ H \ {0}. These graphs form a fairly large class of Cayley sum graphs. Among cases which have been considered previously are the prime sum graphs, in the case where H = pG for some prime number p. In this paper we present their structure and a detailed analysis of their properties. We also consider the simpler graph Γ + G,H , which we refer to as the extended subgroup sum graph, in which x and y are joined if x + y ∈ H: the subgroup sum is obtained by removing from this graph the partial matching of edges having the form {x, −x} when 2x = 0. We study perfectness, clique number and independence number, connectedness, diameter, spectrum, and domination number of these graphs and their complements. We interpret our general results in detail in the prime sum graphs.

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