Spectral Fluctuations of Quantum Graphs

We prove the Bohigas-Giannoni-Schmit conjecture in its most general form for completely connected simple graphs with incommensurate bond lengths. We show that for graphs that are classically mixing (i.e., graphs for which the spectrum of the classical Perron-Frobenius operator possesses a finite gap), the generating functions for all (P, Q) correlation functions for both closed and open graphs coincide (in the limit of infinite graph size) with the corresponding expressions of random-matrix theory, both for orthogonal and for unitary symmetry.