Quantum graph vertices with minimal number of passbands

We study a set of scattering matrices of quantum graphs containing minimal number of passbands, i.e., maximal number of zero elements. The cases of even and odd vertex degree are considered. Using a solution of inverse scattering problem, we reconstruct boundary conditions of scale-invariant vertex couplings. Potential-controlled universal flat filtering properties are found for considered types of vertex couplings. Obtained boundary conditions are approximated by simple graphs carrying only $\delta$ potentials and inner magnetic field.