  We evaluate the variance of coefficients of the characteristic polynomial for
binary quantum graphs using a dynamical approach. This is the first example
where a spectral statistic can be evaluated in terms of periodic orbits for a
system with chaotic classical dynamics without taking the semiclassical limit,
which here is the limit of large graphs. The variance depends on the sizes of
particular sets of primitive pseudo orbits (sets of distinct primitive periodic
orbits): the set of primitive pseudo orbits without self-intersections and the
sets of primitive pseudo orbits with a fixed number of self-intersections, all
of which consist of two arcs of the pseudo orbit crossing at a single vertex.
To show other pseudo orbits do not contribute we give two arguments. The first
is based on a reduction of the variance formula from a sum over pairs of
primitive pseudo orbits to a sum over pseudo orbits where no bonds are
repeated. The second employs a parity argument for the Lyndon decomposition of
words. For families of binary graphs, in the semiclassical limit, we show the
pseudo orbit formula approaches a universal constant independent of the
coefficient of the polynomial. This is obtained by counting the total number of
primitive pseudo orbits of a given length.
