Inverse scattering problem for quantum graph vertices

We demonstrate how the inverse scattering problem of a quantum star graph can be solved by means of diagonalization of Hermitian unitary matrix when the vertex coupling is of the scale invariant (or F\"ul\H{o}p-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.