Upper Eigenvalue Bounds for the Kirchhoff Laplacian on Embbeded Metric Graphs

TL;DR

This paper establishes upper bounds for the eigenvalues of the Kirchhoff Laplacian on compact metric graphs, relating them to the graph's genus, and improves bounds for planar graphs using a spectral correspondence with weighted combinatorial Laplacians.

Contribution

It introduces new upper bounds for Kirchhoff Laplacian eigenvalues based on genus and develops estimates for weighted combinatorial Laplacian eigenvalues, enhancing spectral analysis tools.

Findings

Eigenvalue bounds depend on the graph's genus g.

Bounds are improved for planar graphs where g=0.

New estimates for weighted combinatorial Laplacian eigenvalues.

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 $g = 0$, i.e. if the metric graph is planar. Our results are based on a spectral correspondence between the Kirchhoff Laplacian and a particular a certain 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 weighted case.