  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.
