  We evaluate the variance of coefficients of the characteristic polynomial of
the quantum evolution operator for chaotic 4-regular quantum graphs (networks)
via periodic orbits without taking the semiclassical limit. The variance of the
n-th coefficient is precisely determined by the number of primitive pseudo
orbits (sets of distinct primitive periodic orbits) with n bonds that fall in
the following classes: those with no self-intersections, and those where all
the self-intersections consist of two sections of the pseudo orbit crossing at
a single vertex (2-encounters of length zero).
