Related papers: Equiangular lines and eigenvalue multiplicities

We establish new bounds on the minimum number of distinct eigenvalues among real symmetric matrices with nonzero off-diagonal pattern described by the edges of a graph and apply these to determine the minimum number of distinct eigenvalues…

We determine the maximum number of maximal independent sets of arbitrary graphs in terms of their covering numbers and we completely characterize the extremal graphs. As an application, we give a similar result for K\"onig-Egerv\'ary graphs…

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

We give some new advances in the research of the maximum number of triangles that we may obtain in a simple arrangements of n lines or pseudo-lines.

In 1973, Lemmens and Seidel posed the problem of determining the maximum number of equiangular lines in $\mathbb{R}^r$ with angle $\arccos(\alpha)$ and gave a partial answer in the regime $r \leq 1/\alpha^2 - 2$. At the other extreme where…

In the broad range of studies related to quantum graphs, quantum graph spectra appear as a topic of special interest. They are important in the context of diffusion type problems posed on metric graphs. Theoretical findings suggest that…

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.

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…

Motivated by the linear time algorithm that locates the eigenvalues of a cograph G [10], we investigate the multiplicity of eigenvalue for \lambda \neq -1,0. For cographs with balanced cotrees we determine explicitly the highest value for…

In this note, we study the maximum number $N_\alpha(d)$ of a system of equiangular lines in $\mathbb{R}^d$ with cosine $\alpha$, where $\frac{1}{\alpha}$ is not an odd positive integer. This note is motivated by a remark in a $2018$ paper…

We determine all graphs whose adjacency matrix has at most two eigenvalues (multiplicities included) different from $\pm 1$ and decide which of these graphs are determined by their spectrum. This includes the so-called friendship graphs,…

We present a new approach to compute selected eigenvalues and eigenvectors of the two-parameter eigenvalue problem. Our method requires computing generalized eigenvalue problems of the same size as the matrices of the initial two-parameter…

Let $G$ be an undirected graph on $n$ vertices and let $S(G)$ be the set of all $n \times n$ real symmetric matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of $G$. The inverse eigenvalue…

We consider the eigenvalue problem for the case where the input matrix is symmetric and its entries perturb in some given intervals. We present a characterization of some of the exact boundary points, which allows us to introduce an inner…

We derive the limiting distribution for the largest eigenvalues of the adjacency matrix for a stochastic blockmodel graph when the number of vertices tends to infinity. We show that, in the limit, these eigenvalues are jointly multivariate…

We prove conditions for equality between the extreme eigenvalues of a matrix and its quotient. In particular, we give a lower bound on the largest singular value of a matrix and generalize a result of Finck and Grohmann about the largest…

Spectral graph theory is a captivating area of graph theory that employs the eigenvalues and eigenvectors of matrices associated with graphs to study them. In this paper, we present a collection of $20$ topics in spectral graph theory,…

We give an upper bound on the maximal eigenvalue of the adjacency matrix of a connected graph in terms of its maximum degree, diameter and order. This bound is best possible up to a constant factor and improves prevoius results of…

This article provides sharp bounds for the maximum number of edges possible in a simple graph with restricted values of two of the three parameters, namely, maxi- mum matching size, independence number and maximum degree. We also construct…

There are several notions of duality between lines and points. In this note, it is shown that all these can be studied in a unified way. Most interesting properties are independent of specific choices. It is also shown that either dual…