  We report on high-precision measurements that were performed with
superconducting waveguide networks with the geometry of a tetrahedral and a
honeycomb graph. They consist of junctions of valency three that connect
straight rectangular waveguides of incommensurable lengths. The experiments
were performed in the frequency range of a single transversal mode, where the
associated Helmholtz equation is effectively one dimensional and waveguide
networks may serve as models of quantum graphs with the joints and waveguides
corresponding to the vertices and bonds. The tetrahedral network comprises T
junctions, while the honeycomb network exclusively consists of Y junctions,
that join waveguides with relative angles 90 degree and 120 degree,
respectively. We demonstrate that the vertex scattering matrix, which describes
the propagation of the modes through the junctions strongly depends on
frequency and is non-symmetric at a T junction and thus differs from that of a
quantum graph with Neumann boundary conditions at the vertices. On the
contrary, at a Y junction, similarity can be achieved in a certain frequeny
range. We investigate the spectral properties of closed waveguide networks and
fluctuation properties of the scattering matrix of open ones and find good
agreement with random matrix theory predictions for the honeycomb waveguide
graph.
