  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.
