We present the results of experimental and theoretical study of irregular, tetrahedral microwave networks consisting of coaxial cables (annular waveguides) connected by $T$ joints. The spectra of the networks were measured in the frequency range $0.0001–16\text{GHz}$ in order to obtain their statistical properties such as the integrated nearest neighbor spacing distribution and the spectral rigidity ${\Delta }_3\left(L\right)$. The comparison of our experimental and theoretical results shows that microwave networks can simulate quantum graphs with time reversal symmetry. In particular, we use the spectra of the microwave networks to study the periodic orbits of the simulated quantum graphs. We also present experimental study of directional microwave networks consisting of coaxial cables and Faraday isolators for which the time reversal symmetry is broken. In this case our experimental results indicate that spectral statistics of directional microwave networks deviate from predictions of Gaussian orthogonal ensembles in random matrix theory approaching, especially for small eigenfrequency spacing $s$, results for Gaussian unitary ensembles. Experimental results are supported by the theoretical analysis of directional graphs.

We present the results of the experimental study of the two-port scattering matrix S\ifmmode \hat{}\else \^{}\fi{} elastic enhancement factor WS,$\beta${} for microwave irregular networks simulating quantum graphs with preserved and broken time reversal symmetry in the presence of moderate and strong absorption. In the experiment, quantum graphs with preserved time reversal symmetry were simulated by microwave networks which were built of coaxial cables and attenuators connected by joints. Absorption in the networks was controlled by the length of microwave cables and the use of microwave attenuators. In order to simulate quantum graphs with broken time reversal symmetry we used the microwave networks with microwave circulators. We show that the experimental results obtained for networks with moderate and strong absorption are in good agreement with the ones obtained within the framework of random matrix theory.

The famous question of Kac ``can one hear the shape of a drum?'' addressing the unique connection between the shape of a planar region and the spectrum of the corresponding Laplace operator, can be legitimately extended to scattering systems. In the modified version, one asks whether the geometry of a vibrating system can be determined by scattering experiments. We present the first experimental approach to this problem in the case of microwave graphs (networks) simulating quantum graphs. Our experimental results strongly indicate a negative answer. To demonstrate this we consider scattering from a pair of isospectral microwave networks consisting of vertices connected by microwave coaxial cables and extended to scattering systems by connecting leads to infinity to form isoscattering networks. We show that the amplitudes and phases of the determinants of the scattering matrices of such networks are the same within the experimental uncertainties. Furthermore, we demonstrate that the scattering matrices of the networks are conjugated by the so-called transplantation relation.

We present experimental studies of the power spectrum and other fluctuation properties in the spectra of microwave networks simulating chaotic quantum graphs with violated time reversal invariance. On the basis of our data sets, we demonstrate that the power spectrum in combination with other long-range and also short-range spectral fluctuations provides a powerful tool for the identification of the symmetries and the determination of the fraction of missing levels. Such a procedure is indispensable for the evaluation of the fluctuation properties in the spectra of real physical systems like, e.g., nuclei or molecules, where one has to deal with the problem of missing levels.

We present the results of an experimental and numerical study of the distribution of the reflection coefficient $P\left(R\right)$ and the distributions of the imaginary $P\left(v\right)$ and the real $P\left(u\right)$ parts of the Wigner reaction $K$ matrix for irregular fully connected hexagon networks (graphs) in the presence of strong absorption. In the experiment we used microwave networks, which were built of coaxial cables and attenuators connected by joints. In the numerical calculations experimental networks were described by quantum fully connected hexagon graphs. The presence of absorption introduced by attenuators was modeled by optical potentials. The distribution of the reflection coefficient $P\left(R\right)$ and the distributions of the reaction $K$ matrix were obtained from measurements and numerical calculations of the scattering matrix $S$ of the networks and graphs, respectively. We show that the experimental and numerical results are in good agreement with the exact analytic ones obtained within the framework of random matrix theory.

We use tetrahedral microwave networks consisting of coaxial cables and attenuators connected by T-joints to make an experimental study of Wigner's reaction K matrix for irregular graphs in the presence of absorption. From measurements of the scattering matrix S for each realization of the microwave network we obtain distributions of the imaginary and real parts of K. Our experimental results are in good agreement with theoretical predictions.