Invertibility of adjacency matrices for random d-regular graphs

Abstract

Let d≥3 be a fixed integer, and let A be the adjacency matrix of a random d-regular directed or undirected graph on n vertices. We show that there exists a constant d>0 such that P(A is singular in R)≤n−d, for n sufficiently large. This answers an open problem by Frieze and Vu. The key idea is to study the singularity probability over a finite field Fp. The proof combines a local central limit theorem and a large deviation estimate.