Energy and Laplacian energy of the relative squared graph for symmetric group of order six

The energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix and the Laplacian energy is the sum of the absolute differences between the Laplacian eigenvalues and the average degree of the graph. These spectral measures have found extensive use in algebraic graph theory in the study of commutativity and other structural properties of groups, but very little has been done to the higher-order commutativity relations, like those of relative squared graphs. This paper builds the relative squared graph of the symmetric group of order six and calculates its graph energy and Laplacian energy to study the manifestation of higher- order commutativity by spectral properties. It is proposed to use the methodology as follows the relative squared graph is built with the condition (xy)²=(yx)² and the adjacency and Laplacian matrices are constructed to compute the eigenvalues of these matrices as a measure of the energy. The findings reveal that various subgroup structures of the symmetric group give different values of energy and Laplacian energy which represents different degrees of connectivity and irregularity of the graph. The results of these studies confirm that spectral measures like graph energy, Laplacian energy are useful in the analysis of the structural and commutative behaviour of non-abelian groups using graphs.