On Euler-Sombor Energy of Graphs

TL;DR

This paper introduces the Euler-Sombor (ES) index and matrix for graphs, explores their eigenvalues, and analyzes the ES energy for specific graph classes, expanding spectral graph theory with a new molecular descriptor.

Contribution

It defines the ES matrix and eigenvalues for graphs, and investigates their properties and energy for certain graph classes, providing new insights into spectral graph analysis.

Findings

Computed ES eigenvalues for various graph classes

Analyzed the ES energy and its bounds

Extended spectral graph theory with a new molecular descriptor

Abstract

In 2024, Gutman et al. \cite{I.Gutman 3} defined a new molecular descriptor called as The Euler-Sombor $(ES)$ index of graph. By using this index we define the Euler-Sombor $(ES)$ matrix of a graph $G$ whoes $(i,j)^{th}$ entry is $\sqrt{{d_i}^2+{d_j}^2+d_i.d_j}$ if vertex $v_i$ is adjacent to vertex $v_j$, otherwise $0$. The $ES$ eigenvalues of the graph $G$ are the eigenvalues of its $ES$ matrix ,$ES(G)$. In this paper we discus $ES$ eigenvalues $\nu_i$ and energy of some classes of graphs.