三角条带格上的平衡生成树
Balanced Spanning Trees for Triangular Strip Lattices
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- Rice University（莱斯大学）
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中文总结 AI 辅助
本文针对三角条带格，通过递推关系和渐近分析，研究了平衡生成树的计数及其比例，并给出了均匀随机选取时平衡生成树的极限概率。
中文摘要 AI 辅助
平衡生成树是指一棵生成树，其中存在一条边，其移除后能将顶点划分为两个大小完全相等的连通子树。本文针对$2 \times n$三角条带格（通过在$2 \times n$网格图的每个方格中添加一条对角边得到）建立了生成树数量的显式递推关系，推广了Raff [Raf08]引入的组合计数技术。随后，我们调整了Gallagher和Tapp [GT25]的论证方法，以计算任意三角条带格上的平衡生成树数量。我们建立了当$n \rightarrow \infty$时平衡生成树比例的尖锐渐近界。最后，我们确定了当$n \rightarrow \infty$时，从$2 \times n$三角条带格中均匀随机选取的生成树为平衡生成树的概率。
英文摘要
A balanced spanning tree is a spanning tree that contains an edge whose removal partitions the vertices into exactly two connected subtrees of equal size. In this paper, we establish explicit recurrence relations for the number of spanning trees in $2 \times n$ triangular strip lattices- obtained by adding a diagonal edge to each square of a $2 \times n$ grid graph- generalizing combinatorial counting techniques introduced by Raff [Raf08]. We then adapt arguments of Gallagher and Tapp [GT25] to count balanced spanning trees of arbitrary triangular strip lattices. We establish sharp asymptotic bounds for the proportion of balanced spanning trees as $n \rightarrow \infty$. Finally, we determine the probability that a spanning tree of a $2 \times n$ triangular strip lattice chosen uniformly at random is balanced as $n \rightarrow \infty$.
