Energy and laplacian energy of graphs related to a family of finite groups

Abstract

the energy of a graph is the sum of the absolute value of the eigenvalues of the adjacency matrix of the graph. this quantity is studied in the context of spectral graph theory. the energy of graph was first defined by gutman in 1978. however, the motivation for the study of the energy comes from chemistry, dating back to the work by hukel in the 1930s, where it is used to approximate the total n-electron energy of molecules. recently, the energy of the graph has become an area of interest to many mathematicians and several variations have been introduced. in this research, new theoretical results on the energy and the laplacian energy of some graphs associated to three types of finite groups, which are dihedral groups, generalized quaternion groups and quasidihedral groups are presented. the main aim of this research is to find the energy and laplacian energy of these graphs by using the eigenvalues and the laplacian eigenvalues of the graphs respectively. the results in this research revealed more properties and classifications of dihedral groups, generalized quaternion groups and quasidihedral groups in terms of conjugacy classes of the elements of the groups. the general formulas for the energy and laplacian energy of the conjugacy class graph of dihedral groups, generalized quaternion groups and quasidihedral groups are introduced by using the properties of conjugacy classes of finite groups and the concepts of a complete graph. moreover, the general formula for the energy of the non-commuting graph of these three types of groups are introduced by using some group theory concepts and the properties of the complete multipartite graph. furthermore, the formulas for the laplacian spectrum of the non-commuting graph of dihedral groups, generalized quaternion groups and quasidihedral groups are also introduced, where the proof of the formulas comes from the concepts of an ac-group and the complement of the graph. graphs associated to the relative commutativity degree of subgroups of some dihedral groups are found as the complete multipartite graphs. some formulas for the characteristic polynomials of the adjacency matrices of the graph associated to the relative commutativity degree of subgroups of some dihedral groups and its generalization for some cases depending on the divisors are presented. finally, the energy and the laplacian energy of these graphs for some dihedral groups are computed.