**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.