Related papers: Graphs and Hermitian matrices: discrepancy and sin…

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…

In this note, we use eigenvalue interlacing to derive an inequality between the maximum degree of a graph and its maximum and minimum adjacency eigenvalues. The case of equality is fully characterized.

A mixed multigraph is obtained from an undirected multigraph by orienting a subset of its edges. In this paper, we study a new Hermitian matrix representation of mixed multigraphs, give an introduction to cospectral operations on mixed…

An eigenvalue of the adjacency matrix of a graph is said to be \emph{main} if the all-1 vector is not orthogonal to the associated eigenspace. In this work, we approach the main eigenvalues of some graphs. The graphs with exactly two main…

The inverse eigenvalue problem of a graph $G$ studies the possible spectra of matrices associated with $G$, including as an important subproblem the possible nullities of such a matrix. Much research in this area to date has focused only on…

The spectral properties of signed directed graphs, which may be naturally obtained by assigning a sign to each edge of a directed graph, have received substantially less attention than those of their undirected and/or unsigned counterparts.…

A hollow matrix described by a graph $G$ is a real symmetric matrix having all diagonal entries equal to zero and with the off-diagonal entries governed by the adjacencies in $G$. For a given graph $G$, the determination of all possible…

For the eigenvalues of $p$ complex hermitian $n\times n$ matrices coupled in a chain, we give a method of calculating the spacing functions. This is a generalization of the one matrix case which has been known for a long time.

Consider a graph on the non-singular matrices over a finite field, in which two distinct non-singular matrices are joined by an edge whenever their sum is singular. We prove an upper bound for the independence number of this graph. As a…

In this paper, we establish several new inequalities for twice differantiable mappings that are connected with the celebrated Hermite-Hadamard integral inequality. Some applications for special means of real numbers are also provided.

The goal of this expository note is to give a short, self-contained proof of nearly optimal lower bounds for the second largest eigenvalue of the adjacency matrix of regular graphs.

We expand upon a graph theoretic set of uncertainty principles with tight bounds for difference estimators acting simultaneously in the graph domain and the frequency domain. We show that the eigenfunctions of a modified graph Laplacian and…

The degree matrix of a graph is the diagonal matrix with diagonal entries equal to the degrees of the vertices of $X$. If $X_1$ and $X_2$ are graphs with respective adjacency matrices $A_1$ and $A_2$ and degree matrices $D_1$ and $D_2$, we…

This contribution gives an extensive study on spectra of mixed graphs via its Hermitian adjacency matrix of the second kind { ($N$-matrix for short)} introduced by Mohar \cite{0001}. This matrix is indexed by the vertices of the mixed…

In this paper we discuss some relations between the eigenvalues and the diagonal entries of Hermitian matrices.

We establish a Harnack inequality for finite connected graphs with non-negative Ricci curvature. As a consequence, we derive an eigenvalue lower bound, extending previous results for Ricci flat graphs.

This paper explores interlacing inequalities in the Laplacian spectrum of signed cycles and investigates interlacing relationship between the spectrum of the net-Laplacian of a signed graph and its subgraph formed by removing a vertex…

We give some new bounds for the clique and independence numbers of a graph in terms of its eigenvalues.

A graph is said to be equimatchable if all its maximal matchings are of the same size. In this work we introduce two extensions of the property of equimatchability by defining two new graph parameters that measure how far a graph is from…

We study random graphs with arbitrary distributions of expected degree and derive expressions for the spectra of their adjacency and modularity matrices. We give a complete prescription for calculating the spectra that is exact in the limit…