Related papers: On the eigenvalues of weighted directed graphs
The paper describes an explicit combinatorial formula for a harmonic vector for the Laplacian of a directed graph with arbitrary edge weights. This result was motivated by questions from mathematical economics, and the formula plays a…
We present recent advances in harmonic analysis on infinite graphs. Our approach combines combinatorial tools with new results from the theory of unbounded Hermitian operators in Hilbert space, geometry, boundary constructions, and spectral…
We consider a non self-adjoint Laplacian on a directed graph with non symmetric weights on edges. We give a criterion for the m-accretiveness and the m-sectoriality of this Laplacian. Our results are based on a comparison of this operator…
In this paper we introduce and study generally non-self-adjoint realizations of the Dirac operator on an arbitrary finite metric graph. Employing the robust boundary triple framework, we derive, in particular, a variant of the Birman…
Nonlinear spectral graph theory is an extension of the traditional (linear) spectral graph theory and studies relationships between spectral properties of nonlinear operators defined on a graph and topological properties of the graph…
We discuss Neumann problems for self-adjoint Laplacians on (possibly infinite) graphs. Under the assumption that the heat semigroup is ultracontractive we discuss the unique solvability for non-empty subgraphs with respect to the vertex…
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…
In this note we elaborate on some notions of surface area for discrete graphs which are closely related to the inverse degree. These notions then naturally lead to associated connectivity measures of graphs and to the definition of a…
An important facet of the inverse eigenvalue problem for graphs is to determine the minimum number of distinct eigenvalues of a particular graph. We resolve this question for the join of a connected graph with a path. We then focus on…
We consider two-point non-self-adjoint boundary eigenvalue problems for linear matrix differential operators. The coefficient matrices in the differential expressions and the matrix boundary conditions are assumed to depend analytically on…
We establish a connection between the stability of an eigenvalue under a magnetic perturbation and the number of zeros of the corresponding eigenfunction. Namely, we consider an eigenfunction of discrete Laplacian on a graph and count the…
In this work we introduce a concept of complexity for undirected graphs in terms of the spectral analysis of the Laplacian operator defined by the incidence matrix of the graph. Precisely, we compute the norm of the vector of eigenvalues of…
We study the spectral theory of a class of piecewise centrosymmetric Jacobi operators defined on an associated family of substitution graphs. Given a finite centrosymmetric matrix viewed as a weight matrix on a finite directed path graph…
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.
There are two main notions of a Laplacian operator associated with graphs: discrete graph Laplacians and continuous Laplacians on metric graphs (widely known as quantum graphs). Both objects have a venerable history as they are related to…
In this expository paper we answer two fundamental questions concerning discrete magnetic Schr\"odinger operator associated with weighted graphs. We discuss when formal expressions of such operators give rise to self-adjoint operators,…
We establish several new relations between the discrete transition operator, the continuous Laplacian and the averaging operator associated with combinatorial and metric graphs. It is shown that these operators can be expressed through each…
This paper focuses on certain finite dimensional point derivations for the non-selfadjoint operator algebras corresponding to directed graphs. We begin by analyzing the derivations corresponding to full matrix representations of the tensor…
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…
The Laplacian eigenvalues of a network play an important role in the analysis of many structural and dynamical network problems. In this paper, we study the relationship between the eigenvalue spectrum of the normalized Laplacian matrix and…