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.