Quantum graph vertices with permutation-symmetric scattering probabilities

► There exist various types of couplings in quantum graph vertices. ► The “scale invariant” and the “permutation-symmetric” coupling family are prominent. ► We find a common generalization of scale invariant and permutation-symmetric family. ► It allows us to find equally-transmitting vertex couplings of given characteristics. ► Some of them can be identified as generalized delta and delta-prime couplings.