  The paper contains a simplified and improved version of the results obtained
by the authors earlier. Wave propagation is discussed in a network of branched
thin wave guides when the thickness vanishes and the wave guides shrink to a
one dimensional graph. It is shown that asymptotically one can describe the
propagating waves, the spectrum and the resolvent in terms of solutions of
ordinary differential equations on the limiting graph. The vertices of the
graph correspond to junctions of the wave guides. In order to determine the
solutions of the ODE on the graph uniquely, one needs to know the gluing
conditions (GC) on the vertices of the graph.
  Unlike other publications on this topic, we consider the situation when the
spectral parameter is greater than the threshold, i.e., the propagation of
waves is possible in cylindrical parts of the network. We show that the GC in
this case can be expressed in terms of the scattering matrices related to
individual junctions. The results are extended to the values of the spectral
parameter below the threshold and around it.
