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Mathematical Problems in Engineering
Volume 2013, Article ID 720854, 4 pages
http://dx.doi.org/10.1155/2013/720854
Research Article

Sharp Upper Bounds for the Laplacian Spectral Radius of Graphs

1Department of Mathematics, Shaoyang University, Hunan 422000, China
2Shaoyang Radio & TV University, Hunan 422000, China

Received 15 August 2013; Accepted 10 October 2013

Academic Editor: Miguel A. F. Sanjuán

Copyright © 2013 Houqing Zhou and Youzhuan Xu. 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

The spectrum of the Laplacian matrix of a network plays a key role in a wide range of dynamical problems associated with the network, from transient stability analysis of power network to distributed control of formations. Let be a simple connected graph on vertices and let be the largest Laplacian eigenvalue (i.e., the spectral radius) of . In this paper, by using the Cauchy-Schwarz inequality, we show that the upper bounds for the Laplacian spectral radius of .