The rank of diluted random graphs

Abstract

We investigate the rank of the adjacency matrix of large diluted random graphs: for a sequence of graphs (G_n)_n≥0 converging locally to a Galton–Watson tree T (GWT), we provide an explicit formula for the asymptotic multiplicity of the eigenvalue 0 in terms of the degree generating function φ_∗ of T. In the first part, we show that the adjacency operator associated with T is always self-adjoint; we analyze the associated spectral measure at the root and characterize the distribution of its atomic mass at 0. In the second part, we establish a sufficient condition on φ_∗ for the expectation of this atomic mass to be precisely the normalized limit of the dimension of the kernel of the adjacency matrices of (G_n)_n≥0. Our proofs borrow ideas from analysis of algorithms, functional analysis, random matrix theory and statistical physics.