Upper eigenvalue bounds for the Kirchhoff Laplacian on embedded metric graphs
Marvin Plümer
FernUniversität Hagen, Germany
![Upper eigenvalue bounds for the Kirchhoff Laplacian on embedded metric graphs cover](/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-jst-volume-11-issue-4.png&w=3840&q=90)
Abstract
We derive upper bounds for the eigenvalues of the Kirchhoff Laplacian on a compact metric graph depending on the graph’s genus . These bounds can be further improved if , 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.
Cite this article
Marvin Plümer, Upper eigenvalue bounds for the Kirchhoff Laplacian on embedded metric graphs. J. Spectr. Theory 11 (2021), no. 4, pp. 1857–1894
DOI 10.4171/JST/388