Spectral distributions of adjacency and Laplacian matrices of random graphs
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.
Keywords
Cite
@article{arxiv.1011.2608,
title = {Spectral distributions of adjacency and Laplacian matrices of random graphs},
author = {Xue Ding and Tiefeng Jiang},
journal= {arXiv preprint arXiv:1011.2608},
year = {2010}
}
Comments
Published in at http://dx.doi.org/10.1214/10-AAP677 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)