Related papers: Volume growth and spectrum for general graph Lapla…
We discuss optimal lower bounds for eigenvalues of Laplacians on weighted graphs. These bounds are formulated in terms of the geometry and, more specifically, the inradius of subsets of the graph. In particular, we study the first non-zero…
In this note, we study Laplacians on graphs for which connectivity within certain subgraphs tends to infinity. Our main focus are graphs sharing a common node set on which edge weights within certain clusters grow to infinity. As…
In this paper we characterize emptiness of the essential spectrum of the Laplacian under a hyperbolicity assumption for general graphs. Moreover we present a characterization for emptiness of the essential spectrum for planar tessellations…
In this paper we generalise the results on eigenvalues and eigenvectors of unnormalized (combinatorial) Laplacian of two-dimensional grid presented by Edwards:2013 first to a grid graph of any dimension, and second also to other types of…
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…
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…
Laplace operators on finite compact metric graphs are considered under the assumption that matching conditions at graph vertices are of $\delta$ and $\delta'$ types. Assuming rational independence of edge lengths, necessary and sufficient…
We study the interplay between spectrum, geometry and boundary conditions for two distinguished self-adjoint realisations of the Laplacian on infinite metric graphs, the so-called riedrichs and Neumann extensions. We introduce a new…
In this paper, we prove sharp $\ell^2$ decay estimates of nonnegative generalized subharmonic functions on graphs with positive Laplacian spectrum, which extends the result by Li and Wang on Riemannian manifolds.
We study the volume growth of metric balls as a function of the radius in discrete spaces, and focus on the relationship between volume growth and discrete curvature. We improve volume growth bounds under a lower bound on the so-called…
Graphs with diverse structural characteristics play a central role in modelling and optimization tasks. The ability to generate different types of graphs that exhibit shared properties is likewise essential for algorithm selection and…
In this paper we study the spectral gap of the path graph and illustrate an interesting effect which has been described recently in the continuous setting. More explicitly, in the large-volume limit and in the presence of a certain external…
We examine diagonal combs, a recently identified class of infinite metric graphs whose properties depend on one parameter. These graphs exhibit a fascinating regime where they possess infinite volume while maintaining purely discrete…
We derive several upper bounds on the spectral gap of the Laplacian with standard or Dirichlet vertex conditions on compact metric graphs. In particular, we obtain estimates based on the length of a shortest cycle (girth), diameter, total…
We generalize the weak Omori-Yau maximum principle to the setting of strongly local Dirichlet forms. As an application, we obtain an analytic approach to compare the stochastic completeness of a weighted graph with that of an associated…
We shall prove that under some volume growth condition, the essential spectrum of the Laplacian contains the interval $[(n-1)^2K/4, \infty)$ if an $n$-dimensional Riemannian manifold has an end and the average of the part of the Ricci…
We discuss Laplacians on graphs in a framework of regular Dirichlet forms. We focus on phenomena related to unboundedness of the Laplacians. This includes (failure of) essential selfadjointness, absence of essential spectrum and stochastic…
In this paper, we study extremal values for the determinant of the weighted graph Laplacian under simple nondegeneracy conditions on the weights. We derive necessary and sufficient conditions for the determinant of the Laplacian to be…
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,…
Graph embeddings, a class of dimensionality reduction techniques designed for relational data, have proven useful in exploring and modeling network structure. Most dimensionality reduction methods allow out-of-sample extensions, by which an…