Spectral statistics for quantum graphs: periodic orbits and combinatorics

We consider :the Schrodinger operator on graphs and study the spectral stat istics of a unitary operator which represents the quantum evolution, or a q uantum map on the graph. This operator is the quantum analogue of the class ical evolution operator of the corresponding classical dynamics on the same graph. We derive a trace formula, which expresses the spectral density of the quantum operator in terms of periodic orbits on the graph, and show tha t one can reduce the computation of the two-point spectral correlation func tion to a well defined combinatorial problem. We illustrate this approach b y considering an ensemble of simple graphs. We prove by a direct computatio n that the two-point correlation function coincides with the circular unita ry ensemble expression for 2 x 2 matrices. We derive the same result using the periodic orbit approach in its combinatorial guise. This involves the u se of advanced combinatorial techniques which we explain.