  We consider a family of compact manifolds which shrinks with respect to an
appropriate parameter to a graph. The main result is that the spectrum of the
Laplace-Beltrami operator converges to the spectrum of the (differential)
Laplacian on the graph with Kirchhoff boundary conditions at the vertices. On
the other hand, if the the shrinking at the vertex parts of the manifold is
sufficiently slower comparing to that of the edge parts, the limiting spectrum
corresponds to decoupled edges with Dirichlet boundary conditions at the
endpoints. At the borderline between the two regimes we have a third
possibility when the limiting spectrum can be described by a nontrivial
coupling at the vertices.
