Energy and Laplacian Energy of the Relative co-Prime Graph for Symmetric Group of Order Six

Abstract

The energy of a graph is calculated as the sum of the modulus of eigenvalues of an adjacency matrix for a graph. The Laplacian energy refers to the eigenvalues of the graph’s Laplacian matrix. In graph theory, the co-prime graph of a group is a graph where the vertices represent the elements of a group, and two distinct vertices are connected if and only if their orders are relatively prime. This study is then extended to the relative co-prime graph of a group, where the vertices are the elements of the group, and two distinct vertices are adjacent if and only if their orders are co-prime and any of them is in the subgroup. By using the definition of the relative co-prime graph of a group with respect to a subgroup, some graphs of symmetric group of order six are constructed. Some energies and Laplacian energies of the relative co-prime graph related to the symmetric group of order six are also obtained.