Related papers: The inverse eigenvalue problem of a graph: Multipl…

Let $\mathbb{F}$ be an infinite field with characteristic different from two. For a graph $G=(V,E)$ with $V={1,...,n}$, let $S(G;\mathbb{F})$ be the set of all symmetric $n\times n$ matrices $A=[a_{i,j}]$ over $\mathbb{F}$ with…

Let $\Gamma=(G,\sigma)$ be a signed graph, where $\sigma$ is the sign function on the edges of $G$. The adjacency matrix of $\Gamma=(G, \sigma)$ is a square matrix $A(\Gamma)=A(G, \sigma)=\left(a_{i j}^{\sigma}\right)$, where $a_{i…

The inertia of a graph $G$ is defined to be the triplet $In(G) = (p(G), n(G), $ $\eta(G))$, where $p(G)$, $n(G)$ and $\eta(G)$ are the numbers of positive, negative and zero eigenvalues (including multiplicities) of the adjacency matrix…

The interior transmission eigenvalue problem (ITP) plays a central role in inverse scattering theory and in the spectral analysis of inhomogeneous media. Despite its smooth dependence on the refractive index at the PDE level, the…

Harary and Schwenk posed the problem forty years ago: Which graphs have distinct adjacency eigenvalues? In this paper, we obtain a necessary and sufficient condition for an Hermitian matrix with simple spectral radius and distinct…

We use the line digraph construction to associate an orthogonal matrix with each graph. From this orthogonal matrix, we derive two further matrices. The spectrum of each of these three matrices is considered as a graph invariant. For the…

A general treatment of the spectral problem of quantum graphs and tight-binding models in finite Hilbert spaces is given. The direct spectral problem and the inverse spectral problem are written in terms of simple algebraic equations…

Typically, graph structures are represented by one of three different matrices: the adjacency matrix, the unnormalised and the normalised graph Laplacian matrices. The spectral (eigenvalue) properties of these different matrices are…

Identifying the collection of scalars that represent a non-negative matrix's eigenvalues is known as the non-negative inverse eigenvalue problem (NIEP). Conditions for the existence of a non-negative matrix with a certain spectrum are…

We determine all graphs for which the adjacency matrix has at most two eigenvalues (multiplicities included) not equal to $-2$, or $0$, and determine which of these graphs are determined by their adjacency spectrum.

For signed graphs we provide a cubic polynomial upper bound on the multiplicity of its eigenvalues. We show that this bound is sharp by providing examples of signed graphs in which it is attained. We also discuss particular cases in which…

We review the properties of eigenvectors for the graph Laplacian matrix, aiming at predicting a specific eigenvalue/vector from the geometry of the graph. After considering classical graphs for which the spectrum is known, we focus on…

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…

In a signed graph $G$, an induced subgraph is called a negative clique if it is a complete graph and all of its edges are negative. In this paper, we give the characteristic polynomials and the eigenvalues of some signed graphs having…

The dominating set problem (DSP) is one of the most famous problems in combinatorial optimization. It is defined as follows. For a given simple graph $G=(V,E)$, a dominating set of $G$ is a subset $S\subseteq V$ such that every vertex in $…

In this paper, we describe some recent spectral Moore theorems related to determining the maximum order of a connected graph of given valency and second eigenvalue. We show how these spectral Moore theorems have applications in Alon-Boppana…

The \textit{eccentricity matrix} $\mathcal{E}(G)$ of a connected graph $G$ is obtained from the distance matrix of $G$ by keeping the largest non-zero entries in each row and each column, and leaving zeros in the remaining ones. The…

We investigate the spectrum of the non-backtracking matrix of a graph. In particular, we show how to obtain eigenvectors of the non-backtracking matrix in terms of eigenvectors of a smaller matrix. Furthermore, we find an expression for the…

We investigate properties of signed graphs that have few distinct eigenvalues together with a symmetric spectrum. Our main contribution is to determine all signed $(0,2)$-graphs with vertex degree at most $6$ that have precisely two…

We examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs. We find that when the degree sequence of the graph slowly increases to infinity with the number of…