Spectral distributions of adjacency and Laplacian matrices of random graphs

TL;DR

This paper analyzes the spectral properties of adjacency and Laplacian matrices of random graphs, establishing laws of large numbers, eigenvalue density results, and convergence to well-known spectral distributions.

Contribution

It provides new theoretical results on the spectral distributions and eigenvalue behavior of random graph matrices, including convergence to free convolutions and semi-circular laws.

Findings

Spectral norms and largest eigenvalues follow a law of large numbers.

Normalized largest eigenvalues of Laplacian matrices are dense in a compact interval.

Eigenvalue distributions converge to free convolution and semi-circular laws.

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.