On Some Topological Indices Defined via the Modified Sombor Matrix

Abstract

Let G be a simple graph with the vertex set V={v1,…,vn} and denote by dvi the degree of the vertex vi. The modified Sombor index of G is the addition of the numbers (dvi2+dvj2)-1/2 over all of the edges vivj of G. The modified Sombor matrix AMS(G) of G is the n by n matrix such that its (i,j)-entry is equal to (dvi2+dvj2)-1/2 when vi and vj are adjacent and 0 otherwise. The modified Sombor spectral radius of G is the largest number among all of the eigenvalues of AMS(G). The sum of the absolute eigenvalues of AMS(G) 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 2; 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.

Keywords: Sombor index; adjacency matrix; correlation; modified Sombor energy; modified Sombor matrix.

Figure 1

Double star-type graph $DS(16, 3, 4)$ on 16 vertices and the chain graph $CG\left(10\right)$ on 10 vertices.

Figure 2

Three pairs of equienergetic chemical graphs of the order of at most 7, namely $\{C_4, K_{1,3}\}$, $\{G_1, G_2\}$, and $\{G_3, G_4\}$. Among these three pairs, $\{C_4, K_{1,3}\}$ and $\{G_3,G_4\}$ are neither Sombor equienergetic graphs nor modified Sombor equienergetic graphs.

Figure 3

The scattering of $Bp$ (boiling points) with each of the topological indices ${}^{m}SO$ and ${\mathcal{E}}_{MS}$ for the linear, logarithmic, and quadratic regressions along with the regression equations and $R^2$ (coefficient of determination).