# Bled'11 - 7th Slovenian International Conference on Graph Theory

19-25 June 2011
Bled, Slovenia
Europe/Ljubljana timezone
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# Laplacian Energy for special classes of graphs

Presented by Dr. Renata DEL-VECCHIO
Type: Oral presentation
Track: Graph Spectra and its Applications

## Content

Recently a new concept, the Laplacian energy of a graph, has been defined (,  ) as the sum of the absolute values of the difference between the eigenvalues of the Laplacian matrix and the average degree of the vertices of G. It is the analogue of energy for the Laplacian matrix of G. Let G be a connected graph with n vertices and m edges, the Laplacian Energy of G is then LE(G)=∑_(i=1)^n|μ_i-2m/n| where [μ_1,μ_2,…,μ_n ]is the decreasing sequence of Laplacian eigenvalues of G. A threshold graph is a graph free of P4, C4 or 2K2, and can be obtained by a recursive process which starts with an isolated vertex and where, at each step, either a new isolated vertex or a vertex adjacent to all previous vertices is added (a dominating vertex). In this work we deal only with connected threshold graphs. We define an special class of threshold graphs and we analyse the variation of the Laplacian energy in this family. An split graph is a graph free of C4, C5 or 2K2. Every threshold graph is split, but the converse is not true. We define a family of split non-threshold Laplacian integral graphs, and analyse the Laplacian energy for this graphs comparing them with the Laplacian energy of a special threshold, with the same number of vértices. References;  I. Gutman, B. Zhou, Laplacian energy of a graph, Lin. Algebra Appl. 414 (2006) 29-37.  B. Zhou, I. Gutman, On Laplacian energy of graphs, MATCH Commun. Math. Comput. Chem. 57 (2007) 211-220.

## Place

Location: Bled, Slovenia
Address: Best Western Hotel Kompas Bled

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