  Following the derivation of the trace formulae in the first paper in this
series, we establish here a connection between the spectral statistics of
random regular graphs and the predictions of Random Matrix Theory (RMT). This
follows from the known Poisson distribution of cycle counts in regular graphs,
in the limit that the cycle periods are kept constant and the number of
vertices increases indefinitely. The result is analogous to the so called
"diagonal approximation" in Quantum Chaos. We also show that by assuming that
the spectral correlations are given by RMT to all orders, we can compute the
leading deviations from the Poisson distribution for cycle counts. We provide
numerical evidence which supports this conjecture.
