Eigenvalue bracketing for discrete and metric graphs

TL;DR

This paper introduces eigenvalue estimates for Laplacians on discrete and metric graphs, establishing a correspondence between their spectra and applying the results to covering graphs with spectral gaps.

Contribution

It develops a method to transfer eigenvalue estimates from metric to discrete graphs using boundary conditions, including in exceptional spectral cases.

Findings

Eigenvalue estimates are established for both discrete and metric graph Laplacians.

A spectral correspondence between equilateral metric graphs and discrete graphs is demonstrated.

Examples show the existence of spectral gaps in covering graph Laplacians.

Abstract

We develop eigenvalue estimates for the Laplacians on discrete and metric graphs using different types of boundary conditions at the vertices of the metric graph. Via an explicit correspondence of the equilateral metric and discrete graph spectrum (also in the ``exceptional'' values of the metric graph corresponding to the Dirichlet spectrum) we carry over these estimates from the metric graph Laplacian to the discrete case. We apply the results to covering graphs and present examples where the covering graph Laplacians have spectral gaps.