An algebraic approach to asymptotics of the number of unlabelled bicolored graphs

We define and study two structures associated to permutation groups: Dirichlet characters on permutation groups, and the "cycle form, " a bilinear form on the group algebras of permutation groups. We use Dirichlet characters and the cycle form to find a new upper bound on the number of unlabelled bicolored graphs with p red vertices and q blue vertices. We use this bound to calculate the asymptotic growth rate of the number of such graphs as p, q, answering a 1973 question of Harrison in the case where q-p is fixed. As an application, we show that, in an asymptotic sense, "most" elements of the power set P (\ 1, , p\ \ 1, , q\) are in free ₚ q-orbits.