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We demonstrate how the inverse scattering problem of a quantum star graph canbe solved by means of diagonalization of Hermitian unitary matrix when thevertex coupling is of the scale invariant (or F\"ul\H{o}p-Tsutsui) form. Thisenables the construction of quantum graphs with desired properties in atailor-made fashion. The procedure is illustrated on the example of quantumvertices with equal transmission probabilities.

Inverse scattering problem for quantum graph vertices