On a new variant Sombor matrix and energy of total transformation graphs of regular graphs

Abstract

In this paper, we study the spectral properties of a new variant of Sombor matrix $\mathcal{NS}o(\G)$ for simple graph $\G$. The $\mathcal{NS}o(\G)$ is defined as, for $i\neq j$ its $(i,j)$-entry is equal to $\sqrt{d_{i}^{2}+d_{j}^{2}}$ and 0 otherwise, where $d_{i}$ represents the degree of $i^{th}$ vertex. Explicit expressions for the spectrum, the spectral radius $\eta_{1}$ and the Sombor energy $E_{\mathcal{NS}o}(\G)$ are obtained for total transformation graphs $\G^{xyz}$ of regular graphs using the relation $E_{\mathcal{NS}o}(\G)=2\eta_{1}$. Also as an application, a set of 26 hetero atoms with total $\pi$-electron energy is analyzed, and a regression analysis is performed using $E_{\mathcal{NS}o}(\G)$ for the hetero atoms. The linear regression model yields a correlation coefficient of $r=0.982$.