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