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 $p,q\to \mathrm{\infty }$, 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 $P(\{1,\dots ,p\}\times \{1,\dots ,q\})$ are in free ${\mathrm{\Sigma }}_p\times {\mathrm{\Sigma }}_q$-orbits.