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December 2010 Spectral distributions of adjacency and Laplacian matrices of random graphs
Xue Ding, Tiefeng Jiang
Ann. Appl. Probab. 20(6): 2086-2117 (December 2010). DOI: 10.1214/10-AAP677

Abstract

In this paper, we investigate the spectral properties of the adjacency and the Laplacian matrices of random graphs. We prove that:

 (i) the law of large numbers for the spectral norms and the largest eigenvalues of the adjacency and the Laplacian matrices;

 (ii) under some further independent conditions, the normalized largest eigenvalues of the Laplacian matrices are dense in a compact interval almost surely;

 (iii) the empirical distributions of the eigenvalues of the Laplacian matrices converge weakly to the free convolution of the standard Gaussian distribution and the Wigner’s semi-circular law;

 (iv) the empirical distributions of the eigenvalues of the adjacency matrices converge weakly to the Wigner’s semi-circular law.

Citation

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Xue Ding. Tiefeng Jiang. "Spectral distributions of adjacency and Laplacian matrices of random graphs." Ann. Appl. Probab. 20 (6) 2086 - 2117, December 2010. https://doi.org/10.1214/10-AAP677

Information

Published: December 2010
First available in Project Euclid: 19 October 2010

zbMATH: 1231.05236
MathSciNet: MR2759729
Digital Object Identifier: 10.1214/10-AAP677

Subjects:
Primary: 05C50, 05C80, 15A52, 60B10

Rights: Copyright © 2010 Institute of Mathematical Statistics

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Vol.20 • No. 6 • December 2010
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