Common Neighborhood Energy of Commuting Graphs of Finite Groups

The commuting graph of a finite non-abelian group G with center Z(G), denoted by ?c(G), is simple undirected whose vertex set G?Z(G), and two distinct vertices x y are adjacent if only xy=yx. Alwardi et al. (Bulletin, 2011, 36, 49-59) defined the common neighborhood matrix CN(G) energy Ecn(G) G. A called CN-hyperenergetic Ecn(G)>Ecn(Kn), where n=|V(G)| Kn denotes complete on n vertices. Two graphs H equal number CN-equienergetic Ecn(G)=Ecn(H). In this paper we compute ?c(G) for several classes groups, including class groups such that central quotient isomorphic to symmetries regular polygon, conclude these not CN-hyperenergetic. We shall also obtain some pairs their CN-equienergetic.