Related papers: On isospectral metric graphs
In this article, we develop a perturbative technique to construct families of non-isomorphic discrete graphs which are isospectral for the standard (also called normalised) Laplacian and its signless version. We use vertex contractions as a…
The aim of the present article is to give an overview of spectral theory on metric graphs guided by spectral geometry on discrete graphs and manifolds. We present the basic concept of metric graphs and natural Laplacians acting on it and…
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…
A metrized graph is a compact singular 1-manifold endowed with a metric. A given metrized graph can be modelled by a family of weighted combinatorial graphs. If one chooses a sequence of models from this family such that the vertices become…
According to a recent conjecture, isospectral objects have different nodal count sequences. We study generalized Laplacians on discrete graphs, and use them to construct the first non-trivial counter-examples to this conjecture. In…
In this paper, we study the higher Steklov eigenvalues of graphs on surfaces. We obtain the upper bound of higher Steklov eigenvalues of a finite graph $G$ with boundary $B$ and genus $g$ by using metrical deformation via probability flows.…
Determining and analyzing the spectra of graphs is an important and exciting research topic in theoretical computer science. The eigenvalues of the normalized Laplacian of a graph provide information on its structural properties and also on…
We develop eigenvalue estimates for the Laplacians on discrete and metric graphs using different types of boundary conditions at the vertices of the metric graph. Via an explicit correspondence of the equilateral metric and discrete graph…
The existence of non-isomorphic graphs which share the same Laplace spectrum (to be referred to as isospectral graphs) leads naturally to the following question: What additional information is required in order to resolve isospectral…
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…
Laplacian Eigenvectors of the graph constructed from a data set are used in many spectral manifold learning algorithms such as diffusion maps and spectral clustering. Given a graph constructed from a random sample of a $d$-dimensional…
The Laplacian spread of a graph is the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of the graph. We find that the class of strongly regular graphs attains the maximum of largest…
Upper and lower estimates of eigenvalues of the Laplacian on a metric graph have been established in 2017 by G. Berkolaiko, J.B. Kennedy, P. Kurasov and D. Mugnolo. Both these estimates can be achieved at the same time only by highly…
Laplacian operators on finite compact metric graphs are considered under the assumption that matching conditions at graph vertices are of $\delta$ and $\delta'$ types. An infinite series of trace formulae is obtained which link together two…
In this paper, we obtain a comparison of Steklov eigenvalues and Laplacian eigenvalues on graphs and discuss its rigidity. As applications of the comparison of eigenvalues, we obtain Lichnerowicz-type estimates and some combinatorial…
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…
The rich spectral information of the graph Laplacian has been instrumental in graph theory, machine learning, and graph signal processing for applications such as graph classification, clustering, or eigenmode analysis. Recently, the Hodge…
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…
We introduce a concept of isoperimetric dimension for magnetic graphs, that is, graphs where every edge is assigned a complex number of modulus one. In analogy with the classical case, we show that isoperimetric inequalities imply Sobolev…
We introduce a family of new centralities, the k-spectral centralities. k-Spectral centrality is a measurement of importance with respect to the deformation of the graph Laplacian associated with the graph. Due to this connection,…