The Annals of Applied Probability

Spectral distributions of adjacency and Laplacian matrices of random graphs

Xue Ding and Tiefeng Jiang

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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.

Article information

Ann. Appl. Probab., Volume 20, Number 6 (2010), 2086-2117.

First available in Project Euclid: 19 October 2010

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Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20] 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.) 15A52 60B10: Convergence of probability measures

Random graph random matrix adjacency matrix Laplacian matrix largest eigenvalue spectral distribution semi-circle law free convolution


Ding, Xue; Jiang, Tiefeng. Spectral distributions of adjacency and Laplacian matrices of random graphs. Ann. Appl. Probab. 20 (2010), no. 6, 2086--2117. doi:10.1214/10-AAP677.

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