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We demonstrate that any self-adjoint coupling in a quantum graph vertex canbe approximated by a family of magnetic Schroedinger operators on a tubularnetwork built over the graph. If such a manifold has a boundary, Neumannconditions are imposed at it. The procedure involves a local change of graphtopology in the vicinity of the vertex; the approximation scheme constructed onthe graph is subsequently `lifted' to the manifold. For the correspondingoperator a norm-resolvent convergence is proved, with the naturalidentification map, as the tube diameters tend to zero.

A general approximation of quantum graph vertex couplings by scaled Schroedinger operators on thin branched manifolds