Abstract: We consider a family of open sets $M_\varepsilon$ which
shrinks with respect
to an appropriate parameter $\varepsilon$ to a graph. Under the additional
assumption that the vertex neighbourhoods are small one can show that the
appropriately shifted Dirichlet spectrum of $M_\varepsilon$ converges to
the spectrum of the (differential) Laplacian on the graph with
Dirichlet boundary conditions at the vertices, i.e., a graph
operator \emph{without} coupling between different edges. The
smallness is expressed by a lower bound on the first eigenvalue of a
mixed eigenvalue problem on the vertex neighbourhood. The lower
bound is given by the first transversal mode of the edge
neighbourhood. We also allow curved edges and show that all bounded
eigenvalues converge to the spectrum of a Laplacian acting on the
edge with an additional potential coming from the curvature.
