  Energy level statistics of quantized chaotic systems have been evaluated in
the semiclassical limit via their periodic orbits using the Gutzwiller and
related trace formulae. Here we evaluate a spectral statistic of chaotic
4-regular quantum graphs from their periodic orbits without the semiclassical
limit. The variance of the n-th coefficient of the characteristic polynomial is
determined by the sizes of the sets of distinct primitive periodic orbits with
n bonds which have no self-intersections, and the sizes of the sets with a
given number of self-intersections which all consist of two sections of the
pseudo orbit crossing at a single vertex. Using this result we observe the
mechanism that connects semiclassical results to the total number of orbits
regardless of their structure.
