Related papers: On Eigenvalue Gaps of Integer Matrices

In this paper we obtain bounds for the extreme entries of the principal eigenvector of hypergraphs; these bounds are computed using the spectral radius and some classical parameters such as maximum and minimum degrees. We also study…

We show how positive unital linear maps can be used to obtain lower bounds for the maximum distance between the eigenvalues of two normal matrices. Some related bounds for the spread and condition number of Hermitian matrices are also…

Koolen et al. showed that if a connected graph with smallest eigenvalue at least $-3$ has large minimal valency, then it is $2$-integrable. In this paper, we will prove that a lower bound for the minimal valency is 166.

We study the phenomenon of "crowding" near the largest eigenvalue $\lambda_{\max}$ of random $N \times N$ matrices belonging to the Gaussian Unitary Ensemble (GUE) of random matrix theory. We focus on two distinct quantities: (i) the…

We consider path-connected sets of matrices and the induced paths between eigenvalues. We discuss the equivalence relation generated by these paths, and how it relates to the presence of higher multiplicity eigenvalues realized by the set.…

Let $G$ be a connected graph of order $n$ with diameter $d$. Remoteness $\rho$ of $G$ is the maximum average distance from a vertex to all others and $\partial_1\geq\cdots\geq \partial_n$ are the distance eigenvalues of $G$. In \cite{AH},…

Let $\mathcal{S}$ be the set of all positive-definite, symmetrizable integer matrices with non-zero upper and lower diagonal and $\mathcal{T}$ to be the set of all positive-definite real symmetric matrices with nonzero upper diagonal such…

Let $A$ be an $N\times n$ random matrix whose entries are coordinates of an isotropic log-concave random vector in $\mathbb{R}^{Nn}$. We prove sharp lower tail estimates for the smallest singular value of $A$ in the following cases: (1)…

In this note, we present a systematic method to explicitly compute the determinants and inverses for some generalized Hilbert matrices associated with orthogonal systems with explicit representations. We expressed the determinant, the…

An $L$-matrix is a matrix whose off-diagonal entries belong to a set $L$, and whose diagonal is zero. Let $N(r,L)$ be the maximum size of a square $L$-matrix of rank at most $r$. Many applications of linear algebra in extremal combinatorics…

Let $M_n$ be an $n$ by $n$ random matrix where each entry is +1 or -1 independently with probability 1/2. Our main result implies that the probability that $M_n$ is singular is at most $(1/\sqrt{2} + o(1))^n$, improving on the previous best…

We establish precise right-tail small deviation estimates for the largest eigenvalue of real symmetric and complex Hermitian matrices whose entries are independent random variables with uniformly bounded moments. The proof relies on a Green…

Given a real matrix A with n columns, the problem is to approximate the Gram product AA^T by c << n weighted outer products of columns of A. Necessary and sufficient conditions for the exact computation of AA^T (in exact arithmetic) from c…

We introduce a measure of discrepancy of Hermitian matrices and establish an inequality between the second singular value of a Hermitian matrix and its discrepancy. These results are applied to answer two questions of Fan Chung about graph…

We prove that an n by n random matrix G with independent entries is completely delocalized. Suppose the entries of G have zero means, variances uniformly bounded below, and a uniform tail decay of exponential type. Then with high…

It is known that $N(n)$, the maximum number of mutually orthogonal latin squares of order $n$, satisfies the lower bound $N(n) \ge n^{1/14.8}$ for large $n$. For $h\ge 2$, relatively little is known about the quantity $N(h^n)$, which…

We consider the numbers of positive and negative eigenvalues of matrices of squared distances between randomly sampled i.i.d. points in a given metric measure space. These numbers and their limits, as the number of points grows, in fact…

Much effort has been spent on characterizing the spectrum of the non-backtracking matrix of certain classes of graphs, with special emphasis on the leading eigenvalue or the second eigenvector. Much less attention has been paid to the…

Let $W_n= \frac{1}{\sqrt n} M_n$ be a Wigner matrix whose entries have vanishing third moment, normalized so that the spectrum is concentrated in the interval $[-2,2]$. We prove a concentration bound for $N_I = N_I(W_n)$, the number of…

We examine the adjacency matrices of three-regular graphs representing one-face maps. Numerical studies reveal that the limiting eigenvalue statistics of these matrices are the same as those of much larger, and more widely studied classes…