Transition from a network of thin fibers to the quantum graph: an   explicitly solvable model
  We consider an explicitly solvable model (formulated in the Riemannian geometry terms) for a stationary wave process in a specific thin domain with the Dirichlet boundary conditions on the boundary of the domain. The transition from the solutions of the scattering problem to the solutions of a problem on the limiting quantum graph is studied. We calculate the Lagrangian gluing conditions at vertices for the problem on the limiting graph. If the frequency of the incident wave is above the bottom of the absolutely continuous spectrum, the gluing conditions are formulated in terms of the scattering data of a problem in a neighborhood of each vertex. Near the bottom of the absolutely continuous spectrum the wave propagation is generically suppressed, and the gluing condition is degenerate (any solution of the limiting problem is zero at each vertex). 
S. Molchanov, B. Vainberg