# On the singularity of adjacency matrices for random regular digraphs

@article{Cook2014OnTS,
title={On the singularity of adjacency matrices for random regular digraphs},
author={Nicholas A. Cook},
journal={Probability Theory and Related Fields},
year={2014},
volume={167},
pages={143-200}
}
• Nicholas A. Cook
• Published 2 November 2014
• Mathematics, Computer Science
• Probability Theory and Related Fields
We prove that the (non-symmetric) adjacency matrix of a uniform random d-regular directed graph on n vertices is asymptotically almost surely invertible, assuming $$\min (d,n-d)\ge C\log ^2n$$min(d,n-d)≥Clog2n for a sufficiently large constant $$C>0$$C>0. The proof makes use of a coupling of random regular digraphs formed by “shuffling” the neighborhood of a pair of vertices, as well as concentration results for the distribution of edges, proved in Cook (Random Struct Algorithms. arXiv:1410…
Structure of eigenvectors of random regular digraphs
• Mathematics
Transactions of the American Mathematical Society
• 2019
Let $n$ be a large integer, let $d$ satisfy $C\leq d\leq \exp(c\sqrt{\ln n})$ for some universal constants $c, C>0$, and let $z\in {\mathcal C}$. Further, denote by $M$ the adjacency matrix of a
Invertibility of adjacency matrices for random d-regular graphs
Let $d\geq 3$ be a fixed integer and $A$ be the adjacency matrix of a random $d$-regular directed or undirected graph on $n$ vertices. We show there exist constants $\mathfrak d>0$, \begin{align*}
Circular law for sparse random regular digraphs
• Mathematics
• 2018
Fix a constant $C\geq 1$ and let $d=d(n)$ satisfy $d\leq \ln^{C} n$ for every large integer $n$. Denote by $A_n$ the adjacency matrix of a uniform random directed $d$-regular graph on $n$ vertices.
The circular law for signed random regular digraphs
We consider a large random matrix of the form $Y=A\odot X$, where $A$ the adjacency matrix of a uniform random $d$-regular directed graph on $n$ vertices, with $d=\lfloor p n\rfloor$ for some fixed
The smallest singular value of dense random regular digraphs
• Mathematics
• 2020
Let $A$ be the adjacency matrix of a uniformly random $d$-regular digraph on $n$ vertices, and suppose that $\min(d,n-d)\geq\lambda n$. We show that for any $\kappa \geq 0$,
The circular law for random regular digraphs with random edge weights
We consider random $n\times n$ matrices of the form $Y_n=\frac1{\sqrt{d}}A_n\circ X_n$, where $A_n$ is the adjacency matrix of a uniform random $d$-regular directed graph on $n$ vertices, with
Sharp transition of the invertibility of the adjacency matrices of sparse random graphs
• Mathematics
• 2018
We consider three different models of sparse random graphs:~undirected and directed Erdős-Renyi graphs, and random bipartite graph with an equal number of left and right vertices. For such graphs we
Invertibility of adjacency matrices for random d-regular directed graphs
Let $d\geq 3$ be a fixed integer, and a prime number $p$ such that $\gcd(p,d)=1$. Let $A$ be the adjacency matrix of a random $d$-regular directed graph on $n$ vertices. We show that as a random
The spectral gap of dense random regular graphs
• Mathematics
The Annals of Probability
• 2019
For any $\alpha\in (0,1)$ and any $n^{\alpha}\leq d\leq n/2$, we show that $\lambda(G)\leq C_\alpha \sqrt{d}$ with probability at least $1-\frac{1}{n}$, where $G$ is the uniform random $d$-regular