Upper eigenvalue bounds for the Kirchhoff Laplacian on embedded metric graphs

Abstract

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 D 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.

On torsional rigidity and ground-state energy of compact quantum graphs

“…These weights have extensively discussed in [25,37,51], in the context of spectral theory of quantum graphs. We refer to G = G m,μ constructed above as the weighted combinatorial graph underlying the metric graph G.…”

Impediments to diffusion in quantum graphs: geometry-based upper bounds on the spectral gap

“…(3) If an m-pumpkin P (m ≥ 2) is included in G as an induced subgraph, and if every connected component of G \ P has exactly one point of intersection with P, then (1.4) λ N 2 (G) ≤ 4π 2 s 2 where s denotes the shortest cycle length within P. The first inequality is standard and will be shown in Section 2.2 below; inequality (1.3) has been proved in [25,Theorem 3.11] and depends on the fact that, by Kuratowski's Theorem, a graph is planar if and only if it does not include any subgraph isomorphic to K 5 or K 3,3 (whereas [25,Theorem 4.8] suggests that metric graphs of higher genus have higher λ N 2 ); finally, the proof of inequality (1.4) is based on the principle that attaching pendants to a graph lowers its eigenvalues (see [5,Theorem 3.10]), together with an estimate on the eigenvalues of the pumpkin graph. This inequality also has a counterpart for higher eigenvalues.…”

“…Having seen that the "single metric quantity estimates" of the type given in Theorem 1.3 are rather subtle, we will now present some estimates that use a combination of two metric quantities while having the correct overall scaling (length) −2 . For similar estimates involving other quantities, we refer, e.g., to [1,5,9,13,25,27] and the references therein.…”

Spectral Monotonicity for Schrödinger Operators on Metric Graphs

“…We consciously exclude the case of continuity-Kirchhoff vertex conditions, where λ 1 (H) = 0, but would like to mention that a large body of literature dealing with estimates for the lowest positive eigenvalue (also called spectral gap) as well as its higher eigenvalues exists. We refer the reader to [3,9,14,15,19,32,33,46,47,49,59,60,63,64] and the references therein.…”

“…Many properties of the non-trivial behavior of Schrödinger operators on metric graphs can be investigated in terms of spectral theory, which has been studied in recent years in various perspectives such as spectral estimates, properties of eigenfunctions, inverse problems or questions of isospectrality, see [1,2,9,12,14,18,22,23,24,34,35,39,40,52,57,58,60,61] for a few of the most recent developments.…”