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arXiv · 1011.2608

Spectral distributions of adjacency and Laplacian matrices of random graphs

Xue Ding·Tiefeng Jiang
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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.

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Xue Ding, Tiefeng Jiang. 2010-11-11. Spectral distributions of adjacency and Laplacian matrices of random graphs. https://doi.org/10.1214/10-aap677

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