Interlacing and Friedlander-type inequalities for spectral minimal partitions of metric graphs

TL;DR

This paper establishes inequalities relating spectral minimal energies of metric graphs with Dirichlet and standard Laplacian eigenvalues, extending classical eigenvalue inequalities and providing improved estimates for certain graph classes.

Contribution

It introduces interlacing inequalities for spectral minimal energies of metric graphs, connecting graph topology with eigenvalue estimates, and generalizes Friedlander-type inequalities.

Findings

Derived interlacing inequalities involving Betti number and degree-one vertices.

Established bounds between spectral minimal energies and Laplacian eigenvalues.

Provided improved eigenvalue estimates for specific classes of graphs.

Abstract

We prove interlacing inequalities between spectral minimal energies of metric graphs built on Dirichlet and standard Laplacian eigenvalues, as recently introduced in [Kennedy et al, arXiv:2005.01126]. These inequalities, which involve the first Betti number and the number of degree one vertices of the graph, recall both interlacing and other inequalities for the Laplacian eigenvalues of the whole graph, as well as estimates on the difference between the number of nodal and Neumann domains of the whole graph eigenfunctions. To this end we study carefully the principle of cutting a graph, in particular quantifying the size of a cut as a perturbation of the original graph via the notion of its rank. As a corollary we obtain an inequality between these energies and the actual Dirichlet and standard Laplacian eigenvalues, valid for all compact graphs, which complements a version for tree graphs of Friedlander's inequalities between Dirichlet and Neumann eigenvalues of a domain. In some cases this results in better Laplacian eigenvalue estimates than those obtained previously via more direct methods.