Euler Sombor Energy of a Graph

Abstract

Ivan Gutman introduced the concept of the Euler Sombor index in the chemical graph theory and defined as $EU(G)= \sum_{\substack{pq\in E(G)}}\sqrt{d_p^2+d_q^2+d_pd_q}$. This paper presents the Euler Sombor matrix and defines a new variant of graph energy called Euler Somber energy $E_{ES}(G)$. An interesting finding is that some spectral properties of the Euler Sombor matrix, including its energy, are related to the sum of the second Zagreb index and forgotten topological index. We also obtained Euler Sombor energy of some standard graphs, bounds on the largest eigenvalue and Euler Somber energy. We correlated the values of $E_{ES}(G)$ of some hetero molecular system with their $\pi$-electron energies, we obtained a good correlation with the correlation coefficient $r = 0.9825$.