Eigenvalue bracketing for discrete and metric graphs

28 pages, 6 figures.-- MSC2000 codes: 05C50, 05C70, 47A10.-- ArXiv pre-print available at: http://arxiv.org/abs/0804.1076MR#: MR2446037 (2010a:47076)Zbl#: Zbl 1152.05044We develop eigenvalue estimates for the Laplacians on discrete and metric graphs using various 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.Publicad