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.
"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