Laplacian Energy of Graphs

In the subject of Chemistry, the total π-electron energy of a chemical-molecular graph has a long-known application . This quantity has been studied for decades, in particular by mathematicians who also specialise in Chemistry. In the last two decades, this quantity has become more apparent in the mathematical literature . I found a paper which was published in 2006 by: “Ivan Gutman” and “Bo Zhou”, titled as: “The Laplacian Energy of a Graph” (see: [2]) . After studying this paper, I noticed a great deal of analogy between the earlier defined energy and the laplacian energy, as well as a number of significant differences.  The main goal of this dissertation is to represent a modified version of the proofs given in [2] for obtaining useful upper and lower-bounds for the laplacian energy. We will be showing the important role that the average vertex-degree plays in the proofs and formalisations rather than the use of the size of the graph (the number of edges).        After studying the preliminaries and being introduced to the energy of a graph, the laplacian energy will be defined and some of the important properties of this quantity will be investigated.