The energy and seidel energy of cayley graphs associated to dihedral, alternating and symmetric groups

Abstract

The energy of a simple graph is defined as the summation of the absolute value of the eigenvalues of the adjacency matrix of the graph. It was motivated by the Huckel Molecular Orbital theory. The theory was used by chemists to estimate the energy associated with n-electron orbitals of molecules which is called as conjugated hydrocarbons. Meanwhile, the Seidel energy is defined as the summation of the absolute value of the eigenvalues of the Seidel matrix of the graph. Besides, a Cayley graph associated to a finite group is defined as a graph where its vertices are the elements of a group and two vertices g and h are joined with an edge if and only if h is equal to the product of x and g for some elements x in the subset X of the group. This research combines the topics in graph theory with group theory, namely on the energy and Seidel energy for Cayley graphs, with some finite groups, namely dihedral groups, alternating groups, and symmetric groups. The results are obtained by finding the isomorphism of the Cayley graphs with respect to the subsets of order one and two, and the generating set associated to the groups. The respected Cayley graphs are found and represented as the union of complete graphs, cycle graphs, and complete bipartite graphs. The obtained graphs are then mapped onto their adjacency matrix and Seidel matrix respectively to obtain the eigenvalues and Seidel eigenvalues of the graphs. Some group theory concepts and properties of special graphs are also used to find the generalizations of the eigenvalues of the Cayley graphs. Finally, the energy and the Seidel energy for the Cayley graphs associated to the dihedral groups, alternating groups, and symmetric groups are obtained by using the eigenvalues and the Seidel eigenvalues of the graphs, respectively. The results show that the Seidel energy of Cayley graphs with respect to subsets of order one associated to the groups are equal to their energy. It is also found that the Seidel energy of Cayley graphs with respect to some subsets of order two and the generating sets associated to the groups are larger than their energy.