On The ES Spectrum and ES Energy of Graph

In 2024, Gutman et al. [19] introduced the molecular descriptor called the Euler–Sombor (ES) index to analyze the graphical structure of a molecule. We define the corresponding Euler–Sombor matrix MES for a graph Γ as follows: The entry at position (i,j)-th is given by √ρ_i^2+ρ_j^2+ρ_iρ_j if the vertices μ_i and μ_j are adjacent, and 0 otherwise, where ρ_i denotes the degree of vertex μ_i. In this paper, we investigate the Euler–Sombor eigenvalues of some classes of graphs, and from these ES eigenvalues, we characterize the graphs. We establish bounds for the spectral radius of the graph and the Euler Sombor index of the graph and identify the graphs that attain these extremal values. We obtain the correlation coefficients of ES energy with some physicochemical properties of octane isomers.