The energy of four graphs of some metacyclic 2-groups

Abstract

Let G be a metacyclic 2-group and gamma_G  is the graph of G. The adjacency matrix of gamma_G is a matrix A=[a_ij]  consisting of 0's and 1's in which the entry a_ij is 1 if there is an edge between the ith and jth vertices and 0 otherwise. The energy of a graph is the sum of all absolute values of the eigenvalues of the adjacency matrix of the graph. In this paper, the energy of commuting graph, non-commuting graph, conjugate graph and conjugacy class graph of metacyclic 2-groups are presented. The results show that the energy of these graphs of the groups must be an even integer.