Laplacian Energy of a Graph with Self-Loops

Abstract

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.

Sombor Energy of a Graph with Self-Loops

“…Let G S be the graph with self-loops, obtained from a simple graph G, by attaching a self-loop to each of its vertices belonging to S, where S ⊆ V (G) with |S| = k. Let X(G S ) and d i (G S ) represent edge set of G S and degree of vertex v i in G S , for 1 ≤ i ≤ n respectively. More on degree-based topological indices, energy of a graph with self-loops and terminologies refer [1,2,4,6,7,[14][15][16].…”

New Bounds on the Energy of Graphs with Self-Loops

“…In [24], the authors obtained graphs such that E (G σ ) = E (G) and 1 ≤ σ ≤ n − 1. More on topological indices with self-loops refer to [3,4,25].…”

On the spectrum, energy and Laplacian energy of graphs with self-loops

“…The graph G is referred to as a singular graph [29] if one of its eigenvalues is zero, which is obvious when det A = 0. If all of a graph's eigenvalues differ from zero, the graph is said to be non singular [5] . Then det A ≠ 0.…”