Nullity and energy of the cubic power graph

Abstract

Let G be a finite abelian group with identity 0, G1 = {3t | t ? G} ? G and G2 = G \ G1. The cubic power graph ?cpg of G is an undirected simple graph with vertex set G, such that two distinct vertices x and y are adjacent in ?cpg if and only if x + y ? G1 \ {0}. In this paper, we first prove certain properties of the cubic power graph ?cpg of finite abelian groups. Finally, we observe some properties on the nullity of the cubic power graph ?cpg and also show that the energy of the cubic power graph of a finite abelian group of order n is bounded below by ?(n ? 3)(n ? 1)/3 if n is divisible by 3 and n ? 1 if n otherwise.