arXiv subjects
Publications and source records attributed to Vilmar Trevisan.
A symmetric matrix $M=(m_{ij}) \in \mathbb{R}^{n \times n}$ is said to be associated with an $n$-vertex graph $G=(V,E)$ with vertex set $\{v_1,\ldots,v_n\}$ if, for every $i \neq j$, we have $m_{ij} \neq 0$ if and only if $\{v_i,v_j\}\in E$. We prove that, for every cograph $G$, there is a matrix $M$ associated with $G$ for which the number of distinct eigenvalues is at most 4.
The Brualdi-Solheid problem asks which graph achieves the extremal (maximum or minimum) spectral radius for a given class of graphs. This paper addresses the Brualdi-Solheid problem for \( \mathcal{G}_{n,\beta} \), the family of graphs with order \( n \) and matching number \( \beta \), aiming to identify its spectrally minimal graphs i.e., those that minimize the spectral radius \(\rho(G)\). We introduce the novel concept of ``quasi-adjacency'' relation, developing a unified structural classification framework for trees in \(\mathcal{G}_{n,\beta}\), which clarifies structural properties and provides a constructive method to generate trees with fixed \(\beta\). By showing that all spectrally minimal graphs in \( \mathcal{G}_{n,\beta} \) are trees, we further narrow the search for extremal graphs. Additionally, we apply this framework to the representative cases \(\beta=2,3,4\), obtaining the minimizers by explicit structural formulas involving parameters related to \(n\).
This paper investigates spectral properties of the deformed Laplacian matrix, which merges the Laplacian and signless Laplacian matrices of a graph through a one-parameter family of matrices. We present general results on the eigenvalues of these matrices for simple undirected graphs. Additionally, we analyze the spectrum of the deformed Laplacian in the specific cases of trees and H-join graphs. For trees, we derive strong results on the localization of eigenvalues, while for H-join graphs, we explicitly compute the spectrum of the deformed Laplacian.
We study limit points of the spectral radii of Laplacian matrices of graphs. We adapted the method used by J. B. Shearer in 1989, devised to prove the density of adjacency limit points of caterpillars, to Laplacian limit points. We show that this fails, in the sense that there is an interval for which the method produces no limit points. Then we generalize the method to Laplacian limit points of linear trees and prove that it generates a larger set of limit points. The results of this manuscript may provide important tools for proving the density of Laplacian limit points in $[4.38+, \infty)$.
We study limit points of the spectral radii of $A_α$-matrices of graphs. Adapting a method used by J. B. Shearer in 1989, we prove a density property of $A_α$-limit points of caterpillars for $α$ close to zero. Precisely, we show that for $α\in [0, 1/2)$ there exists a positive number $τ_2(α)>2$ such that any value $λ> τ_2(α)$ is an $A_α$-limit point. We also determine the existence of other intervals for which all its points are $A_α$-limit points.
In this note, we present a structural description of certain connected cographs having $k \geq 2$ main signless Laplacian eigenvalues. This result allows us to characterize the cographs which are quasi-threshold graphs with two main $\mathbf{Q}$-eigenvalues. In addition, we describe all the quasi-threshold graphs belonging to the subclass of generalized core-satellite graphs with $k \geq 2$ main $\mathbf{Q}$-eigenvalues.
We show that for threshold graphs, the eigenvalues of the signless Laplacian matrix interlace with the degrees of the vertices. As an application, we show that the signless Brouwer conjecture holds for threshold graphs, i.e., for threshold graphs the sum of the k largest eigenvalues is bounded by the number of edges plus k + 1 choose 2.
Given a digraph D, the complementarity spectrum of the digraph is defined as the set of complementarity eigenvalues of its adjacency matrix. This complementarity spectrum has been shown to be useful in several fields, particularly in spectral graph theory. The differences between the properties of the complementarity spectrum for (undirected) graphs and for digraphs, makes the study of the latter of particular interest, and characterizing strongly connected digraphs with a small number of complementarity eigenvalues is a non trivial problem. Recently, strongly connected digraphs with one and two complementarity eigenvalues have been completely characterized. In this paper we study strongly connected digraphs with exactly three elements in the complementarity spectrum, ending with a complete characterization. This leads to a structural characterization of general digraphs having three complementarity eigenvalues.
Given a tree $T$, let $q(T)$ be the minimum number of distinct eigenvalues in a symmetric matrix whose underlying graph is $T$. It is well known that $q(T)\geq d(T)+1$, where $d(T)$ is the diameter of $T$, and a tree $T$ is said to be diminimal if $q(T)=d(T)+1$. In this paper, we present families of diminimal trees of any fixed diameter. Our proof is constructive, allowing us to compute, for any diminimal tree $T$ of diameter $d$ in these families, a symmetric matrix $M$ with underlying graph $T$ whose spectrum has exactly $d+1$ distinct eigenvalues.
Let $M=(m_{ij})$ be a symmetric matrix of order $n$ whose elements lie in an arbitrary field $\mathbb{F}$, and let $G$ be the graph with vertex set $\{1,\ldots,n\}$ such that distinct vertices $i$ and $j$ are adjacent if and only if $m_{ij} \neq 0$. We introduce a dynamic programming algorithm that finds a diagonal matrix that is congruent to $M$. If $G$ is given with a tree decomposition $\mathcal{T}$ of width $k$, then this can be done in time $O(k|\mathcal{T}| + k^2 n)$, where $|\mathcal{T}|$ denotes the number of nodes in $\mathcal{T}$. Among other things, this allows one to compute the determinant, the rank and the inertia of a symmetric matrix in time $O(k|\mathcal{T}| + k^2 n)$.
In this paper we study the complementarity spectrum of digraphs, with special attention to the problem of digraph characterization through this complementarity spectrum. That is, whether two non-isomorphic digraphs with the same number of vertices can have the same complementarity eigenvalues. The complementarity eigenvalues of matrices, also called Pareto eigenvalues, has led to the study of the complementarity spectrum of (undirected) graphs and, in particular, the characterization of undirected graphs through these eigenvalues is an open problem. We characterize the digraphs with one and two complementarity eigenvalues, and we give examples of non-isomorphic digraphs with the same complementarity spectrum.
In this paper, we define and obtain several properties of the (adjacency) energy of a hypergraph. In particular, bounds for this energy are obtained as functions of structural and spectral parameters, such as Zagreb index and spectral radius. We also study how the energy of a hypergraph varies when a vertex/edge is removed or when an edge is divied. In addition, we solved the extremal problem energy for the class of hyperstars, and show that the energy of a hypergraph is never an odd number.
We study a general class of recurrence relations that appear in the application of a matrix diagonalization procedure. We find general closed formula and determine analytical properties of the solutions. We finally apply these findings in several problems involving eigenvalues of graphs.
We show that the number of Laplacian eigenvalues greater than the average degree of a tree having $n$ vertices is at most $\lfloor\frac{n}{2} \rfloor$.
We address the problem of ordering trees with the same degree sequence by their spectral radii. To achieve that, we consider 2-switch transformations which preserve the degree sequence and establish when the index decreases. Our main contribution is to determine a total ordering of a particular family by their indices according to a given parameter related to sizes in the tree.
In this paper, we study energies associated with hypergraphs. More precisely, we obtain results for the incidence and the singless Laplacian energies of uniform hypergraphs. In particular, we obtain bounds for the incidence energy as functions of well known parameters, such as maximum degree, Zagreb index and spectral radius. We also relate the incidence and signless Laplacian energies of a hypergraph with the adjacency energies of its subdivision graph and line multigraph, respectively. In addition, we compute the signless Laplacian energy for the class of the power hypergraphs.
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 inequalities involving the ratio and difference between the two extreme entries of this vector.
A generalized power hypergraph $\mathcal{H}^k_s$ is obtained from a base hypergraph $\mathcal{H}$ by means of some simple edge-expansion operations. Kang, Liu, Qi and Yuan [8] proved that the nonzero eigenvalues of $\mathcal{H}$ give rise to nonzero eigenvalues of $\mathcal{H}^k_s$. In this paper we show that all nonzero eigenvalues of $\mathcal{H}^k_s$ may be computed from the eigenvalues of its base hypergraph $\mathcal{H}$ and of its subgraphs. To prove this, we derive spectral results about edge-expansion operations that may be interesting on their own sake.