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We discuss resonances for Schrdinger operators on metric graphs which consists of a finite compact part and a finite number of halflines attached to it; the vertex coupling is assumed to be of the 6-type or certain modifications of it. Using exterior complex scaling on the graph we show that the resolvent and scattering resonances coincide in this case.

We derive asymptotic expansion for the spectrum of Hamiltonians with a strong attractive interaction supported by a smooth surface in R 3, either infinite and asymptotically planar, or compact and closed. Its second term is found to be determined by a Schrdinger type operator with an effective potential expressed in terms of the interaction support curvatures.

We demonstrate how the inverse scattering problem of a quantum star graph can be solved by means of diagonalization of the Hermitian unitary matrix when the vertex coupling is of the scale-invariant (or Flp-Tsutsui) form. This enables the construction of quantum graphs with desired properties in a tailor-made fashion. The procedure is illustrated on the example of quantum vertices with equal transmission probabilities.