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

TL;DR

This paper introduces algebraic tools to analyze the asymptotic behavior of unlabelled bicolored graphs, providing new bounds and answering a longstanding question about their growth rates.

Contribution

It develops a novel algebraic framework using Dirichlet characters and cycle forms to bound and compute asymptotics of unlabelled bicolored graphs.

Findings

Derived a new upper bound on the number of unlabelled bicolored graphs

Calculated the asymptotic growth rate for large p,q

Showed most elements of the power set are in free orbits under permutation groups

Abstract

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 $p,q\rightarrow\infty$, 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 $\Sigma_p\times \Sigma_q$-orbits.