  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.
