Related papers: Sparse Random Matrices have Simple Spectrum

In this note we describe the singular locus of diagonally-dominant Hermitian matrices with nonnegative diagonal entries over the reals, the complex numbers, and the quaternions. This yields explicit expressions for the probability that such…

In 2003, van Dam and Haemers posed a fundamental question in spectral graph theory: does there exist a ``sensible'' matrix whose spectrum determines a random graph up to isomorphism? This paper introduces the class of {\em natural graph…

We compute spectra of symmetric random matrices describing graphs with general modular structure and arbitrary inter- and intra-module degree distributions, subject only to the constraint of finite mean connectivities. We also evaluate…

We develop a theoretical approach to compute the conditioned spectral density of $N \times N$ non-invariant random matrices in the limit $N \rightarrow \infty$. This large deviation observable, defined as the eigenvalue distribution…

Let $A$ be a rectangular matrix of size $m\times n$ and $A_1$ be the random matrix where each entry of $A$ is multiplied by an independent $\{0,1\}$-Bernoulli random variable with parameter $1/2$. This paper is about when, how and why the…

In recent work on equiangular lines, Jiang, Tidor, Yuan, Zhang, and Zhao showed that a connected bounded degree graph has sublinear second eigenvalue multiplicity. More generally they show that there cannot be too many eigenvalues near the…

In the present work we consider off-diagonal Jacobi matrices with uncertainty in the position of sparse perturbations. We prove (Theorem 3.2) that the sequence of Pr\"ufer angles (\theta_{k}^{\omega})_{k\geq 1} is u.d mod \pi for all \phi…

Let ${\mathcal D}_{n,d}$ be the set of all $d$-regular directed graphs on $n$ vertices. Let $G$ be a graph chosen uniformly at random from ${\mathcal D}_{n,d}$ and $M$ be its adjacency matrix. We show that $M$ is invertible with probability…

The inverse eigenvalue problem of a given graph $G$ is to determine all possible spectra of real symmetric matrices whose off-diagonal entries are governed by the adjacencies in $G$. Barrett et al. introduced the Strong Spectral Property…

We consider the empirical eigenvalue distribution of random real symmetric matrices with stochastically independent skew-diagonals and study its limit if the matrix size tends to infinity. We allow correlations between entries on the same…

The paper gives a thorough introduction to spectra of digraphs via its Hermitian adjacency matrix. This matrix is indexed by the vertices of the digraph, and the entry corresponding to an arc from $x$ to $y$ is equal to the complex unity…

In this work, we study some statistical properties of the extreme eigenstates of the randomly-weighted adjacency matrices of random graphs. We focus on two random graph models: Erd\H{o}s-R\'{e}nyi (ER) graphs and random geometric graphs…

Motivated by the intriguing behavior displayed in a dynamic network that models a population of extreme introverts and extroverts (XIE), we consider the spectral properties of ensembles of random split graph adjacency matrices. We discover…

In this article, we study random graphs with a given degree sequence $d_1, d_2, \cdots, d_n$ from the configuration model. We show that under mild assumptions of the degree sequence, the spectral distribution of the normalized Laplacian…

We study the eigenvectors and eigenvalues of random matrices with iid entries. Let $N$ be a random matrix with iid entries which have symmetric distribution. For each unit eigenvector $\mathbf{v}$ of $N$ our main results provide a small…

It is well known that a graph is bipartite if and only if the spectrum of its adjacency matrix is symmetric. In the present paper, this assertion is dissected into three separate matrix results of wider scope, which are extended also to…

Linial-Meshulam complex is a random simplicial complex on $n$ vertices with a complete $(d-1)$-dimensional skeleton and $d$-simplices occurring independently with probability p. Linial-Meshulam complex is one of the most studied…

We study the singularity probability of n*n random matrices with i.i.d. entries from highly biased discrete distributions. We obtain sharp non-asymptotic bounds for this probability and derive estimates on the least singular values. Our…

We consider random non-normal matrices constructed by removing one row and column from samples from Dyson's circular ensembles or samples from the classical compact groups. We develop sparse matrix models whose spectral measures match these…

We study random graphs with possibly different edge probabilities in the challenging sparse regime of bounded expected degrees. Unlike in the dense case, neither the graph adjacency matrix nor its Laplacian concentrate around their…