  We establish an asymptotic relation between the spectrum of the discrete
Laplacian associated to discretizations of a half-translation surface with a
flat unitary vector bundle and the spectrum of the Friedrichs extension of the
Laplacian with von Neumann boundary conditions.
  As an interesting byproduct of our study, we obtain Harnack-type estimates on
"almost harmonic" discrete functions, defined on the graphs, which approximate
a given surface.
  The results of this paper will be later used to relate the asymptotic
expansion of the number of spanning trees, spanning forests and weighted
cycle-rooted spanning forests on the discretizations to the corresponding
zeta-regularized determinants.
