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<p>We obtain first order linear partial differential equations which are satisfied by exponential generating functions of two variables for the number of labeled connected bipartite graphs with given Betti number. By solving these equations inductively, we obtain the explicit form of generating functions and derive the asymptotic behavior of their coefficients. We also introduce a family of basic graphs to classify labeled connected bipartite graphs and give another expression of the generating functions as the sum over basic graphs of rational functions of those for the number of labeled bipartite rooted spanning trees.</p>

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Enumeration of labeled connected bipartite graphs with given Betti number

  • 期刊论文

  • 发表日期：

    2025年9月17日

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  • Taro Hasui,
  • Tomoyuki Shirai,
  • Satoshi Yabuoku

摘要

We obtain first order linear partial differential equations which are satisfied by exponential generating functions of two variables for the number of labeled connected bipartite graphs with given Betti number. By solving these equations inductively, we obtain the explicit form of generating functions and derive the asymptotic behavior of their coefficients. We also introduce a family of basic graphs to classify labeled connected bipartite graphs and give another expression of the generating functions as the sum over basic graphs of rational functions of those for the number of labeled bipartite rooted spanning trees.