Empirical Distributions of Laplacian Matrices of Large Dilute Random Graphs

We study the spectral properties of the Laplacian matrices and the normalized Laplacian matrices of the Erdös-Rényi random graph G(n, pn) for large n. Although the graph is simple, we discover some interesting behaviors of the two Laplacian matrices. In fact, under the dilute case, that is, pn ∈ (0, 1) and npn → ∞, we prove that the empirical distribution of the eigenvalues of the Laplacian matrix converges to a deterministic distribution, which is the free convolution of the semicircle law and N(0, 1). However, for its normalized version, we prove that the empirical distribution converges to the semi-circle law.