In this paper, we investigate the spectral properties of the adjacency and the Laplacian matrices of random graphs. We prove that:
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(i) the law of large numbers for the spectral norms and the largest eigenvalues of the adjacency and the Laplacian matrices;
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(ii) under some further independent conditions, the normalized largest eigenvalues of the Laplacian matrices are dense in a compact interval almost surely;
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(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;
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(iv) the empirical distributions of the eigenvalues of the adjacency matrices converge weakly to the Wigner’s semi-circular law.
Publié le : 2010-12-15
Classification:
Random graph,
random matrix,
adjacency matrix,
Laplacian matrix,
largest eigenvalue,
spectral distribution,
semi-circle law,
free convolution,
05C80,
05C50,
15A52,
60B10
@article{1287494555,
author = {Ding, Xue and Jiang, Tiefeng},
title = {Spectral distributions of adjacency and Laplacian matrices of random graphs},
journal = {Ann. Appl. Probab.},
volume = {20},
number = {1},
year = {2010},
pages = { 2086-2117},
language = {en},
url = {http://dml.mathdoc.fr/item/1287494555}
}
Ding, Xue; Jiang, Tiefeng. Spectral distributions of adjacency and Laplacian matrices of random graphs. Ann. Appl. Probab., Tome 20 (2010) no. 1, pp. 2086-2117. http://gdmltest.u-ga.fr/item/1287494555/