Related papers: Laplacians on Metric Graphs: Eigenvalues, Resolven…
The paper aims to study the spectral properties of elliptic operators with highly inhomogeneous coefficients and related issues concerning wave propagation in high-contrast media. A unified approach to solving problems in bounded domains…
In this paper we consider an initial-boundary value problem related to some network dynamics where the underlying graph has unbounded edges. We show that there exists a C0-semigroup for this problem using a general result from the…
We introduce a semigroup framework for Laplacians on directed hypergraphs, extending the classical heat flow models on graphs and establishing hypergraphs as prototypical models for non-Markovian diffusion. We apply spectral surgery methods…
We establish a sharp lower bound on the first non-trivial eigenvalue of the Laplacian on a metric graph equipped with natural (i.e., continuity and Kirchhoff) vertex conditions in terms of the diameter and the total length of the graph.…
We define a number of natural (from geometric and combinatorial points of view) deformation spaces of valuations on finite graphs, and study functions over these deformation spaces. These functions include both direct metric invariants…
Given i.i.d. observations uniformly distributed on a closed manifold $\mathcal{M}\subseteq \mathbb{R}^p$, we study the spectral properties of the associated empirical graph Laplacian based on a Gaussian kernel. Our main results are…
In this paper, we characterize all graphs with eigenvectors of the signless Laplacian and adjacency matrices with components equal to $\{- 1, 0, 1\}.$ We extend the graph parameter max $k$-cut to square matrices and prove a general sharp…
In this paper, we develop two approaches to investigation of inverse spectral problems for a new class of nonlocal operators on metric graphs. The Laplace differential operator is considered on a star-shaped graph with nonlocal integral…
We provide explicit upper bounds for the eigenvalues of the Laplacian on a finite metric tree subject to standard vertex conditions. The results include estimates depending on the average length of the edges or the diameter. In particular,…
The independence number, coloring number and related parameters are investigated in the setting of oriented hypergraphs using the spectrum of the normalized Laplace operator. For the independence number, both an inertia--like bound and a…
Exploring the relationship between geometry and the resonant frequencies of a shape is of interest to pure and applied mathematicians. These resonant frequencies are related to the spectrum of the Laplacian, a partial differential operator.…
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 paper, we show the existence of a sequence of eigenvalues for a Dirichlet problem involving two mixed fractional operators with different orders. We provide lower and upper bounds for the sum of the eigenvalues. Applications of…
The Laplace-Beltrami operator on (the surface of) a triaxial ellipsoid admits a sequence of real eigenvalues diverging to plus infinity. By introducing ellipsoidal coordinates, this eigenvalue problem for a partial differential operator is…
Negative-index metamaterials possess a negative refractive index and thus present an interesting substance for designing uncommon optical effects such as invisibility cloaking. This paper deals with operators encountered in an…
We establish the existence of analytic curves of eigenvalues for the Laplace-Neumann operator through an analytic variation of the metric of a compact Riemannian manifold $M$ with boundary by means of a new approach rather than Kato's…
We introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several…
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…
We are interested in the spectrum of the Dirichlet Laplacian in thin broken strips with angle $\alpha$. Playing with symmetries, this leads us to investigate spectral problems for the Laplace operator with mixed boundary conditions in…
We consider an arbitrary metric graph, to which we glue another graph with edges of lengths proportional to $\varepsilon$, where $\varepsilon$ is a small positive parameter. On such graph, we consider a general self-adjoint second order…