  We report on experiments that were performed with microwave waveguide systems
and demonstrate that in the frequency range of a single transversal mode they
may serve as a model for closed and open quantum graphs. These consist of bonds
that are connected at vertices. On the bonds, they are governed by the
one-dimensional Schr\"odinger equation with boundary conditions imposed at the
vertices. The resulting transport properties through the vertices may be
expressed in terms of a vertex scattering matrix. Quantum graphs with
incommensurate bond lengths attracted interest within the field of quantum
chaos because, depending on the characteristics of the vertex scattering
matrix, its wave dynamic may exhibit features of a typical quantum system with
chaotic counterpart. In distinction to microwave networks, which serve as an
experimental model of quantum graphs with Neumann boundary conditions, the
vertex scattering matrices associated with a waveguide system depend on the
wavenumber and the wave functions can be determined experimentally. We analyze
the spectral properties of microwave waveguide systems with preserved and
partially violated time-reversal invariance, and the properties of the
associated wave functions. Furthermore, we study properties of the scattering
matrix describing the measurement process within the frame work of random
matrix theory for quantum chaotic scattering systems.
