Kirchhoff Laplacian Research Articles

A geometric construction of isospectral magnetic graphs

We present a geometrical construction of families of finite isospectral graphs labelled by different partitions of a natural number r of given length s (the number of summands). Isospectrality here refers to the discrete magnetic Laplacian with normalised weights (including standard weights). The construction begins with an arbitrary finite graph {varvec{G}} with normalised weight and magnetic potential as a building block from which we construct, in a first step, a family of so-called frame graphs ({varvec{F}}_a)_{a in {mathbb {N}}}. A frame graph {varvec{F}}_a is constructed contracting a copies of G along a subset of vertices V_0. In a second step, for any partition A=(a_1,dots ,a_s) of length s of a natural number r (i.e., r=a_1+dots +a_s) we construct a new graph {varvec{F}}_A contracting now the frames {varvec{F}}_{a_1},dots ,{varvec{F}}_{a_s} selected by A along a proper subset of vertices V_1subset V_0. All the graphs obtained by different s-partitions of rge 4 (for any choice of V_0 and V_1) are isospectral and non-isomorphic. In particular, we obtain increasing finite families of graphs which are isospectral for given r and s for different types of magnetic Laplacians including the standard Laplacian, the signless standard Laplacian, certain kinds of signed Laplacians and, also, for the (unbounded) Kirchhoff Laplacian of the underlying equilateral metric graph. The spectrum of the isospectral graphs is determined by the spectrum of the Laplacian of the building block G and the spectrum for the Laplacian with Dirichlet conditions on the set of vertices V_0 and V_1 with multiplicities determined by the numbers r and s of the partition.

Self-adjoint and Markovian extensions of infinite quantum graphs.

We investigate the relationship between one of the classical notions of boundaries for infinite graphs, graph ends, and self‐adjoint extensions of the minimal Kirchhoff Laplacian on a metric graph. We introduce the notion of finite volume for ends of a metric graph and show that finite volume graph ends is the proper notion of a boundary for Markovian extensions of the Kirchhoff Laplacian. In contrast to manifolds and weighted graphs, this provides a transparent geometric characterization of the uniqueness of Markovian extensions, as well as of the self‐adjointness of the Gaffney Laplacian — the underlying metric graph does not have finite volume ends. If, however, finitely many finite volume ends occur (as is the case of edge graphs of normal, locally finite tessellations or Cayley graphs of amenable finitely generated groups), we provide a complete description of Markovian extensions upon introducing a suitable notion of traces of functions and normal derivatives on the set of graph ends.

Upper eigenvalue bounds for the Kirchhoff Laplacian on embedded metric graphs

We derive upper bounds for the eigenvalues of the Kirchhoff Laplacian on a compact metric graph depending on the graph’s genus g . These bounds can be further improved if g=0 , i.e., if the metric graph is planar. Our results are based on a spectral correspondence between the Kirchhoff Laplacian and a particular combinatorial weighted Laplacian. In order to take advantage of this correspondence, we also prove new estimates for the eigenvalues of the weighted combinatorial Laplacians that were previously known only in the unweighted case.

Graph‐like spaces approximated by discrete graphs and applications

AbstractWe define a distance between energy forms on a graph‐like metric measure space and on a suitable discrete weighted graph using the concept of quasi‐unitary equivalence. We apply this result to metric graphs, graph‐like manifolds (e.g. a small neighbourhood of an embedded metric graph) or pcf self‐similar fractals as metric measure spaces with energy forms associated with canonical Laplacians, e.g., the Kirchhoff Laplacian on a metric graph resp. the (Neumann) Laplacian on a manifold (with boundary), and express the distance of the associated energy forms in terms of geometric quantities. In particular, we show that there is a sequence of domains converging to a pcf self‐similar fractal such that the corresponding (Neumann) energy forms converge to the fractal energy form. As a consequence, the spectra and suitable functions of the associated Laplacians converge, the latter in operator norm.

Existence and non-existence results for fractional Kirchhoff Laplacian problems

Existence and non-existence results for fractional Kirchhoff Laplacian problems

Zero Measure and Singular Continuous Spectra for Quantum Graphs

We introduce a dynamically defined class of unbounded, connected, equilateral metric graphs on which the Kirchhoff Laplacian has zero Lebesgue measure spectrum and a nontrivial singular continuous part. A new local Borg--Marchenko uniqueness result is obtained in order to utilize Kotani theory for aperiodic subshifts satisfying Boshernitzan's condition.

Existence and multiplicity of positive solutions for a class of Kirchhoff Laplacian type problems

This paper deals with the existence and multiplicity of positive solutions for a class of Kirchhoff Laplacian type problems in whole R3. The multiplicity of solutions is established by variational methods and Ljusternik-Schnirelmann theory.

On a nonlocal asymmetric Kirchhoff problem

This work is devoted to study the existence of nontrivial solutions to nonlocal asymmetric problems involving the [Formula: see text]-Laplacian. [Formula: see text] where [Formula: see text] is a bounded domain with smooth boundary, [Formula: see text] is a Kirchhoff function, [Formula: see text] and [Formula: see text] is of subcritical polynomial or subcritical exponential growth. Moreover, the existence of nontrivial solutions for the above problem is obtained by using variational methods combined with the Moser–Trudinger inequality. Our interest then is to study [Formula: see text] without the analogue of Ambrosetti–Rabinowitz superquadratic condition ([Formula: see text] condition for short) in the positive semi-axis. To the best of our best knowledge, our results are new even in the asymmetric Kirchhoff Laplacian and [Formula: see text]-Laplacian cases.

Absence of Absolutely Continuous Spectrum for the Kirchhoff Laplacian on Radial Trees

In this paper we prove that the existence of absolutely continuous spectrum of the Kirchhoff Laplacian on a radial metric tree graph together with a finite complexity of the geometry of the tree implies that the tree is in fact eventually periodic. This complements the results by Breuer and Frank in \cite{BreuerFrank2009} in the discrete case as well as for sparse trees in the metric case.