Related papers: The adjacency matrix and the discrete Laplacian ac…
We give a sufficient condition for the essential self-adjointness of a perturbation of the square of the magnetic Laplacian on an infinite weighted graph. The main result is applicable to graphs whose degree function is not necessarily…
Given a locally finite simple graph so that its degree is not bounded, every self-adjoint realization of the adjacency matrix is unbounded from above. In this note we give an optimal condition to ensure it is also unbounded from below. We…
We give inequalities relating the eigenvalues of the adjacency matrix and the Laplacian of a graph, and its minimum and maximum degrees. The results are applied to derive new conditions for quasi-randomness of graphs.
We study some spectral properties of a matrix that is constructed as a combination of a Laplacian and an adjacency matrix of simple graphs. The matrix considered depends on a positive parameter, as such we consider the implications in…
On finite metric graphs we consider Laplace operators, subject to various classes of non-self-adjoint boundary conditions imposed at graph vertices. We investigate spectral properties, existence of a Riesz basis of projectors and similarity…
In this paper we consider the discrete Laplacian acting on 1-forms and we study its spectrum relative to the spectrum of the 0-form Laplacian. We show that the non zero spectrum can coincide for these Laplacians with the same nature. We…
We study the uniqueness of self-adjoint and Markovian extensions of the Laplacian on weighted graphs. We first show that, for locally finite graphs and a certain family of metrics, completeness of the graph implies uniqueness of these…
We discuss the connections tying Laplacian matrices to abstract duality and planarity of graphs.
The characteristic polynomials of the adjacency matrix of line graphs of caterpillars and then the characteristic polynomials of their Laplacian or signless Laplacian matrices are characterized, using recursive formulas. Furthermore, the…
In this paper, we define the structure of $n$-simplicial complex, we consider generalizations of the Laplacians to simplicial complexes of higher dimension and we develop the notion of $\chi$-completeness for simplicial complexes.…
Let $G$ be a simple graph. In 1986, Herbert Wilf asked what kind of graphs have an eigenvector with entries formed only by $\pm 1$? In this paper, we answer this question for the adjacency, Laplacian and signless Laplacian matrix of a…
We study self-adjoint matrix polynomial equations in a single variable and prove existence of self-adjoint solutions under some assumptions on the leading form. Our main result is that any self-adjoint matrix polynomial equation of odd…
A matching M is a dominating induced matching of a graph, if every edge of the graph is either in $M$ or has a common end-vertex with exactly one edge in $M$. The concept of complete dominating induced matching is introduced as graphs where…
In this paper, we introduce the notion of oriented faces especially triangles in a connected oriented locally finite graph. This framework then permits to define the Laplace operator on this structure of the 2-simplicial complex. We develop…
In this paper we {\em discuss} diverse aspects of mutual relationship between adjoints and formal adjoints of unbounded operators bearing a matrix structure. We emphasize on the behaviour of row and column operators as they turn out to be…
We prove a general essential self-adjointness criterion for sub-Laplacians on complete sub-Riemannian manifolds, defined with respect to singular measures. As a consequence, we show that the intrinsic sub-Laplacian (i.e. defined w.r.t.…
The traditional adjacency matrix of a mixed graph is not symmetric in general, hence its eigenvalues may be not real. To overcome this obstacle, several authors have recently defined and studied various Hermitian adjacency matrices of…
The Laplacian matrix of a simple graph is the difference of the diagonal matrix of vertex degree and the (0,1) adjacency matrix. In the past decades, the Laplacian spectrum has received much more and more attention, since it has been…
We study the Laplacian in a smooth bounded domain, with a varying Robin boundary condition singular at one point. The associated quadratic form is not semi-bounded from below, and the corresponding Laplacian is not self-adjoint, it has the…
We study the relation between approximate joint diagonalization of self-adjoint matrices and the norm of their commutator, and show that almost commuting self-adjoint matrices are almost jointly diagonalizable by a unitary matrix.