Modified Sombor Spectral Radii and Modified Sombor Energies of Splitting and Shadow Graphs
Ahmad Bilal, Mobeen Munir · Scientific Annals of Computer Science · 2024
Gutman et al. introduced Sombor and Modified Somber index because of the rapidly growing applications in chemistry and network analysis. Graph energy ε(G) and the spectral radius ℘(G) of graph G are essential components that are associated with the eigenvalues of the matrix of graph G and, chemically, with the intermolecular forces. These graph invariants have many useful applications in computer sciences, networking, and molecular computing. There are numerous variants of the ε(G) and ℘(G) attained by substituting another matrix in place of an adjacency matrix. Modified Sombor energy, MSε(G) is defined as the sum of absolute eigenvalues of the modified Sombor matrix or in other words we can say that modified Sombor energy, MSε(G) is the trace norm of modified Sombor matrix. Modified Sombor spectral radius, ℘MS is defined as the largest absolute eigenvalue of the modified Sombor matrix. The major focus of this article is on the MSε(G), ℘MS of the generalized shadow and splitting graphs. The only realistic problem in which we are particularly interested in how MSε(Splt(G)) and MSε(Sht(G)) are comparable to MSε(G). On similar lines we are also interested in how ℘MS(Splt(G)) and ℘MS(Sht(G)) are comparable to ℘MS(G). We were able to address these challenges by focusing on splitting and shadow graphs.