Spectral correlations of individual quantum graphs

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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 σ--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.