Eigenvalue bracketing for discrete and metric graphs

Details Eigenvalue bracketing for discrete and metric graphs Eigenvalue bracketing for discrete and metric graphs

2008-04-07

arxiv.org/abs/0804.1076v1

Mathematics

Spectral Theory

32 pages, 6 figures

Scientific paper

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.

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