Parameter-dependent spectral statistics of chaotic quantum graphs: Neumann versus circular orthogonal ensemble boundary conditions.

The parameter-dependent spectral statistics of totally connected quantum graphs with n = 4-30 vertices, such as the parametric velocities correlation functions and the distribution of curvatures, are studied. The inverse participation ratio (IPR), an important measure of localization effects, was also numerically investigated. In the calculations, we successfully used two different theoretical approaches. The first approach was based on the graphs' eigenenergies and wave functions calculations, while the second one used the eigenphases and the eigenvectors of the bond scattering matrix S(k). We considered graphs with Neumann and circular orthogonal ensemble (COE) boundary conditions. We show that in contrast to large Neumann graphs, for which the departure of many parameter-dependent spectral statistics from the random matrix theory (RMT) predictions is observed, for large COE graphs, the spectral statistics and IPR are in good agreement with the RMT predictions.