The scattering matrix relates the vectors of incoming and outgoing plane-wave coefficients at v , c = S ( k ) c ^ ^ . For self-adjoint matching conditions S is unitary. An element of σ σ ( u v ) ( v w ) of S is a complex transition amplitude from a directed edge ( u v ) to the edge ( v w ) which in general depends on k . However, for a large class of matching conditions the S-matrix is independent of k . With Neumann matching conditions for example