  We generalize the scattering approach to quantum graphs to quantum graphs
with with piecewise constant potentials and multiple excitation modes. The free
single-mode case is well-known and leads to the trace formulas of Roth, Kottos
and Smilansky. By introducing an effective reduced scattering picture we are
able to introduce new exact trace formulas in the more general setting. The
latter are derived and discussed in details with some numerical examples for
illustration. Our generalization is motivated by both experimental applications
and fundamental theoretical considerations. The free single-mode quantum graphs
are an extreme idealization of reality that, due to the simplicity of the model
allows to understand a large number of generic or universal phenomena. We lift
some of this idealization by considering the influence of evanescent modes that
only open above threshold energies. How to do this theoretically in a closed
model in general is a challenging question of fundamental theoretical interest
and we achieve this here for quantum graphs.
