Periodic orbit evaluation of a spectral statistic of quantum graphs without the semiclassical limit

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).