On the Spectral Radius and Sombor Energy of the Non-Commuting Graph for Dihedral Groups

Abstract

The non-commuting graph, denoted by , is defined on a finite group , with its vertices are elements of excluding those in the center  of . In this graph, two distinct vertices are adjacent whenever they do not commute in . The graph  can be associated with several matrices including the most basic matrix, which is the adjacency matrix, , and a matrix called Sombor matrix, denoted by . The entries of  are either the square root of the sum of the squares of degrees of two distinct adjacent vertices, or zero otherwise. Consequently, the adjacency and Sombor energies of  is the sum of the absolute eigenvalues of the adjacency and Sombor matrices of , respectively, whereas the spectral radius of  is the maximum absolute eigenvalues. Throughout this paper, we find the spectral radius obtained from the spectrum of  and the Sombor energy of  for dihedral groups of order ,  where . Moreover, there is an almost linear correlation between the Sombor energy and the adjacency energy of  for  which is slightly different than reported earlier in previous literature.