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arXiv · 2407.07870

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

A. Salch
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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 $Σ_p\times Σ_q$-orbits.

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A. Salch. 2024-07-10. An algebraic approach to asymptotics of the number of unlabelled bicolored graphs. https://arxiv.org/abs/2407.07870

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