On Some Topological Indices Defined via the Modified Sombor Matrix

Let G be a simple graph with the vertex set [Formula: see text] and denote by [Formula: see text] the degree of the vertex [Formula: see text]. The modified Sombor index of G is the addition of the numbers [Formula: see text] over all of the edges [Formula: see text] of G. The modified Sombor matrix [Formula: see text] of G is the n by n matrix such that its [Formula: see text]-entry is equal to [Formula: see text] when [Formula: see text] and [Formula: see text] are adjacent and 0 otherwise. The modified Sombor spectral radius of G is the largest number among all of the eigenvalues of [Formula: see text]. The sum of the absolute eigenvalues of [Formula: see text] 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 [Formula: see text]; 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.