2024/07/10 by Andrew Salch, Salch, A.
Computer Science · Mathematics · #05A16 #05C30 #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Markov Chains and Monte Carlo Methods
paper · pdf · doi:10.48550/arxiv.2407.07870
openalex publication_date 2024/07/10 · openalex created_date 2024/07/13 · openalex updated_date 2026/07/28
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 Σp× Σq-orbits.