Quantum graph vertices with permutation-symmetric scattering probabilities

Boundary conditions in quantum graph vertices are generally given in terms of a unitary matrix $U$. Observing that if $U$ has at most two eigenvalues, then the scattering matrix $\mathcal{S}(k)$ of the vertex is a linear combination of the identity matrix and a fixed Hermitian unitary matrix, we construct vertex couplings with this property: For all momenta $k$, the transmission probability from the $j$-th edge to $\mathrm{\ell }$-th edge is independent of $(j,\mathrm{\ell })$, and all the reflection probabilities are equal. We classify these couplings according to their scattering properties, which leads to the concept of generalized $\delta$ and ${\delta }^{\mathrm{\prime }}$ couplings.