Journal of Applied Mathematics
Volume 2012 (2012), Article ID 404067, 11 pages
http://dx.doi.org/10.1155/2012/404067
Research Article

On the Laplacian Coefficients and Laplacian-Like Energy of Unicyclic Graphs with Vertices and Pendent Vertices

Xinying Pai1,2 and Sanyang Liu1

1Department of Mathematics, Xidian University, Shanxi Xi'an 710071, China
2College of Science, China University of Petroleum, Shandong Qingdao 266580, China

Received 12 June 2012; Accepted 13 September 2012

Academic Editor: Alvaro Valencia

Copyright © 2012 Xinying Pai and Sanyang Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let be the characteristic polynomial of the Laplacian matrix of a graph of order . In this paper, we give four transforms on graphs that decrease all Laplacian coefficients and investigate a conjecture A. Ilić and M. Ilić (2009) about the Laplacian coefficients of unicyclic graphs with vertices and pendent vertices. Finally, we determine the graph with the smallest Laplacian-like energy among all the unicyclic graphs with vertices and pendent vertices.