Eigenvalue bracketing for discrete and metric graphs

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.

32 pages, 6 figures

References in corpus (6)

• Spectra of self-adjoint extensions and applications to solvable Schroedinger operators
• Quantum graphs: an introduction and a brief survey
• Spectra of Schroedinger operators on equilateral quantum graphs
• Equilateral quantum graphs and boundary triples
• Spectral analysis of metric graphs and related spaces
• First order approach and index theorems for discrete and metric graphs