Graph-like models for thin waveguides with Robin boundary conditions

We discuss the limit of small width for the Laplacian defined on a waveguide with Robin boundary conditions. Under suitable hypothesis on the scaling of the curvature, we prove the convergence of the Robin Laplacian to the Laplacian on the corresponding graph. We show that the projections on each transverse mode generically give rise to decoupling conditions between the edges of the graph while exceptionally a coupling can occur. The non decoupling conditions are related to the existence of resonances at the thresholds of the continuum spectrum.