Related papers: Laplacians on infinite graphs: discrete vs continu…
We investigate quantum graphs with infinitely many vertices and edges without the common restriction on the geometry of the underlying metric graph that there is a positive lower bound on the lengths of its edges. Our central result is a…
This paper is devoted to the investigation of the spectral theory and dynamical properties of periodic graphs which are not locally finite but carry non-negative, symmetric and summable edge weights. These graphs are shown to exhibit rather…
Motivated by discrete Laplacian differential operators with various accuracy orders in numerical analysis, we introduce new matrices attached to a simple graph that can be considered graph Laplacians with higher accuracy. In particular, we…
We study the differential operator $A=\frac{d^4}{dx^4}$ acting on a connected network $\mathcal{G}$ along with $\mathcal L^2$, the square of the discrete Laplacian acting on a connected discrete graph $\mathsf{G}$. For both operators we…
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…
We study spectral properties of the standard (also called Kirchhoff) Laplacian and the anti-standard (or anti-Kirchhoff) Laplacian on a finite, compact metric graph. We show that the positive eigenvalues of these two operators coincide…
For a finite graph, we establish natural isomorphisms between eigenspaces of a Laplace operator acting on functions on the edges and eigenspaces of a transfer operator acting on functions on one-sided infinite non-backtracking paths.…
Normalized Laplacian matrices of graphs have recently been studied in the context of quantum mechanics as density matrices of quantum systems. Of particular interest is the relationship between quantum physical properties of the density…
We introduce a non-backtracking Laplace operator for graphs and we investigate its spectral properties. With the use of both theoretical and computational techniques, we show that the spectrum of this operator captures several structural…
In the present article magnetic Laplacians on a graph are analyzed. We provide a complete description of the set of all operators which can be obtained from a given self-adjoint Laplacian by perturbing it by magnetic fields. In particular,…
Recently, much of the existing work in manifold learning has been done under the assumption that the data is sampled from a manifold without boundaries and singularities or that the functions of interest are evaluated away from such points.…
We consider metric graphs with a uniform lower bound on the edge lengths but no further restrictions. We discuss how to describe every local self-adjoint Laplace operator on such graphs by boundary conditions in the vertices given by…
This paper deals with spectral graph theory issues related to questions of monotonicity and comparison of eigenvalues. We consider finite directed graphs with non symmetric edge weights and we introduce a special self-adjoint operator as…
We deal with the existence of three distinct solutions for a poly-Laplacian system with a parameter on finite graphs and a $(p,q)$-Laplacian system with a parameter on locally finite graphs.The main tool is an abstract critical points…
We consider Laplacians on periodic both discrete and metric equilateral graphs. Their spectrum consists of an absolutely continuous part (which is a union of non-degenerate spectral bands) and flat bands, i.e., eigenvalues of infinite…
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…
On a periodic planar graph whose edge weights satisfy a certain simple geometric condition, the discrete Laplacian and d-bar operators have the property that their determinants and inverses only depend on the local geometry of the graph. We…
We consider a non self-adjoint Laplacian on a directed graph with non symmetric edge weights. We give necessary conditions for this Laplacian to be sectorial. We introduce a special self-adjoint operator and compare its essential spectrum…
We construct quantum models of two particles on a compact metric graph with singular two-particle interactions. The Hamiltonians are self-adjoint realisations of Laplacians acting on functions defined on pairs of edges in such a way that…
In this paper, we study the graph-theoretic analogues of vector Laplacian (or Helmholtz operator) and vector Laplace equation. We determine the graph matrix representation of vector Laplacian and obtain the dimension of solution space of…