Some graphs of metabelian groups of order 24 and their energy

Abstract

The energy of a graph G is the sum of all absolute values of the eigenvalues of the adjacency matrix. An adjacency matrix is a square matrix where the rows and columns consist of 0 or 1-entry depending on the adjacency of the vertices of the graph. A commuting graph of a group is a graph whose vertex set is the non-central elements of the group and whose edges are pairs of vertices that commute. Meanwhile, a noncommuting graph is a graph whose vertex set is the non-central elements of the group but the edges are the pairs of vertices that do not commute. A conjugacy class graph is a graph with the non-central conjugacy classes vertices. Two vertices are connected if the order of the conjugacy classes have a common prime divisor. Besides, a conjugate graph is a graph whose vertex set is the non-central elements of the group where two distinct vertices are joined if they are conjugate. Furthermore, a group G is said to be metabelian if there exists a normal subgroup H in G such that both H and the factor group G/H are abelian. In this research, the energies of commuting graphs, noncommuting graphs, conjugacy class graphs and conjugate graphs for all nonabelian metabelian group of order 24 are determined. The computations of the graphs and adjacency matrices for the energy of graphs are determined with the assistance of Groups, Algorithms and Programming (GAP) and Maple 2016 softwares. The results show that the energy of graphs of the groups in the study must be an even integer in the case that the energy is rational.