Spectral correlations of individual quantum graphs
  We investigate the spectral properties of chaotic quantum graphs. We demonstrate that the `energy'--average over the spectrum of individual graphs can be traded for the functional average over a supersymmetric non--linear $\sigma$--model action. This proves that spectral correlations of individual quantum graphs behave according to the predictions of Wigner--Dyson random matrix theory. We explore the stability of the universal random matrix behavior with regard to perturbations, and discuss the crossover between different types of symmetries. 
Sven Gnutzmann and Alexander Altland