Invertibility of adjacency matrices for random d-regular graphs

Abstract

Let $d\ge 3$ 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 $\mathfrak{d}>0$ such that

$$\mathbb{P}(A\text{is singular in}\mathbb{R})\le n^{-\mathfrak{d}},$$

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 ${\mathbb{F}}_p$. The proof combines a local central limit theorem and a large deviation estimate.