On Some Topological Indices Defined via the Modified Sombor Matrix.

Abstract

Let G be a simple graph with the vertex set and denote by the degree of the vertex . The modified Sombor index of G is the addition of the numbers over all of the edges of G. The modified Sombor matrix of G is the n by n matrix such that its -entry is equal to when and are adjacent and 0 otherwise. The modified Sombor spectral radius of G is the largest number among all of the eigenvalues of . The sum of the absolute eigenvalues of is known as the modified Sombor energy of G. Two graphs with the same modified Sombor energy are referred to as modified Sombor equienergetic graphs. In this article, several bounds for the modified Sombor index, the modified Sombor spectral radius, and the modified Sombor energy are found, and the corresponding extremal graphs are characterized. By using computer programs (Mathematica and AutographiX), it is found that there exists only one pair of the modified Sombor equienergetic chemical graphs of an order of at most seven. It is proven that the modified Sombor energy of every regular, complete multipartite graph is ; this result gives a large class of the modified Sombor equienergetic graphs. The (linear, logarithmic, and quadratic) regression analyses of the modified Sombor index and the modified Sombor energy together with their classical versions are also performed for the boiling points of the chemical graphs of an order of at most seven.