The purpose of this paper is to extend the concept of Laplacian energy from simple graph to a graph with self-loops. Let G be a simple graph of order n, size m and GS is the graph obtained from G by adding σ self-loops. We define Laplacian energy of GS aswhere µ1(GS), µ2(GS), . . . , µn(GS) are eigenvalues of the Laplacian matrix of GS. In this paper some basic proprties of Laplacian eigenvalues and bounds for Laplacian energy of GS are investigated. This paper is limited to bounds in analogy with bounds of E(G) and LE(G) but with some significant differences, more sharper bounds can be found.

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 GS be a graph with k self-loops obtained from a simple graph G, we define Sombor index for GS as SO(GS)where S ⊆ 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 GS. Also explores the correlation between Sombor energy of GS and the total π−electron energies associated with the corresponding hetero-molecular systems.