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The Scientific World Journal
Volume 2013 (2013), Article ID 472956, 7 pages
http://dx.doi.org/10.1155/2013/472956
Research Article

Bounds of the Spectral Radius and the Nordhaus-Gaddum Type of the Graphs

School of Mathematics and Information Science, Leshan Normal University, Leshan 614004, China

Received 28 February 2013; Accepted 14 May 2013

Academic Editors: H.-L. Liu and Y. Wang

Copyright © 2013 Tianfei Wang et al. 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 Laplacian spectra are the eigenvalues of Laplacian matrix , where and are the diagonal matrix of vertex degrees and the adjacency matrix of a graph , respectively, and the spectral radius of a graph is the largest eigenvalue of . The spectra of the graph and corresponding eigenvalues are closely linked to the molecular stability and related chemical properties. In quantum chemistry, spectral radius of a graph is the maximum energy level of molecules. Therefore, good upper bounds for the spectral radius are conducive to evaluate the energy of molecules. In this paper, we first give several sharp upper bounds on the adjacency spectral radius in terms of some invariants of graphs, such as the vertex degree, the average 2-degree, and the number of the triangles. Then, we give some numerical examples which indicate that the results are better than the mentioned upper bounds in some sense. Finally, an upper bound of the Nordhaus-Gaddum type is obtained for the sum of Laplacian spectral radius of a connected graph and its complement. Moreover, some examples are applied to illustrate that our result is valuable.