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 \(G_S\) is the graph obtained from \(G\) by adding \(\sigma\) self-loops. We define Laplacian energy of \(G_S\) as \(LE(G_{S})=\sum\limits_{i=1}^{n}\left\vert \mu_{i}(G_{S})-\frac{2m+\sigma}{n}\right \vert\) where \(\mu_{1}(G_{S}), \mu_{2}(G_{S}), \dots, \mu_{n}(G_{S})\) are eigenvalues of the Laplacian matrix of \(G_S\). In this paper some basic properties of Laplacian eigenvalues and bounds for Laplacian energy of \(G_S\) 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.
