Laplacian Energy of a Graph with Self-Loops

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 as (Formula Presented) 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.