Abstract
This study aims to extend the notion of degree-based topological index, associated adjacency-type matrix and its energy from a simple graph to a graph with self-loops. Let \(G_S\) be a graph with \(k\) self-loops obtained from a simple graph \(G\), we define Sombor index for \(G_S\) as \(SO(G_S)=\sum\limits_{v_iv_j\in X(G)}\sqrt{d^2_i(G_S)+d^2_j(G_S)}+\sqrt{2}\sum\limits_{v_i\in S}d_i(G_S) \), where \(S\subseteq V(G)\) having self-loop to each of its vertices in \( S \). In addition we investigate some fundamental properties of Sombor eigenvalues, McClelland and Koolen-Moulton-type bound for Sombor energy of \(G_S\). Also explores the correlation between Sombor energy of \(G_S\) and the total \(\pi\)-electron energies associated with the corresponding hetero-molecular systems.
