The rank of random regular digraphs of constant degree

TL;DR

This paper proves that the adjacency matrices of large random directed regular graphs are almost surely of rank at least n-1, combining switchings and eigenvector delocalization techniques.

Contribution

It establishes a high-probability lower bound on the rank of adjacency matrices of random regular digraphs, advancing understanding of their spectral properties.

Findings

Rank of adjacency matrix is at least n-1 with high probability

Uses switchings and eigenvector delocalization methods

Results hold for large fixed degree d

Abstract

Let $d$ be a fixed large integer. For any $n$ larger than $d$, let $A_n$ be the adjacency matrix of the random directed $d$-regular graph on $n$ vertices, with the uniform distribution. We show that $A_n$ has rank at least $n-1$ with probability going to one as $n$ goes to infinity. The proof combines the method of simple switchings and a recent result of the authors on delocalization of eigenvectors of $A_n$.