生成函数与强连通有向图的熵层级
Generating Functions and the Entropy Hierarchy of Strongly Connected Digraphs
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中文总结 AI 辅助
本文用生成函数方法研究强连通有向图的拓扑熵，通过蝴蝶参数化建立熵层级，确定小顶点数的熵序并识别结构分岔，给出极值段与最小阶数公式。
中文摘要 AI 辅助
我们提出了一种生成函数方法来研究有限强连通有向图的拓扑熵，全程处理允许自环但排除多重边的有向图。利用路径生成函数的主奇点，我们通过新的解析组合论证，重新获得了先前在无环谱设置中研究的已知第一和第二正熵极小元，并将相应的极值陈述推广到当前框架。对于由具有m个顶点和m+1条边的强连通有向图组成的类$\mathcal{SC}_{m+1}(m)$，我们引入了一个统一的$(t,k_1,k_2)$-蝴蝶参数化。$\mathcal{B}^{\\,t}_{k_1,k_2}$的熵仅依赖于$(k_1,k_2)$，并由方程$1-z^{k_1}-z^{k_2}=0$的唯一根$R\in(0,1)$通过$h=-\ln R$确定。该参数化在$\mathcal{SC}_{m+1}(m)$内产生了详细的熵层级。我们引入了金字塔熵图，完全确定了$m\leq 7$时的熵序，并识别了其在$m=8$处的第一个结构分岔。我们进一步建立了由十个熵最小值和两个熵最大值组成的最大稳定初始段和终端段，并导出了实现不超过给定阈值的正熵所需的最小阶数的显式公式。
英文摘要
We present a generating-function approach to the topological entropy of finite strongly connected digraphs, working throughout with digraphs in which loops are allowed but multiple edges are excluded. Using dominant singularities of path-generating functions, we recover by new analytic--combinatorial arguments the known first and second positive-entropy minimizers from the previously studied loopless spectral setting, and extend the corresponding extremal statements to the present framework. For the class $\mathcal{SC}_{m+1}(m)$ of strongly connected digraphs with $m$ vertices and $m+1$ edges, we introduce a unified $(t,k_1,k_2)$-butterfly parametrization. The entropy of $\mathcal{B}^{\,t}_{k_1,k_2}$ depends only on $(k_1,k_2)$ and is determined by the unique root $R\in(0,1)$ of $1-z^{k_1}-z^{k_2}=0$ via $h=-\ln R$. This parametrization yields a detailed entropy hierarchy within $\mathcal{SC}_{m+1}(m)$. We introduce the Pyramidal Entropy Diagram, determine the entropy order completely for $m\leq 7$, and identify its first structural bifurcation at $m=8$. We further establish maximal stable initial and terminal segments consisting of ten entropy minima and two entropy maxima, respectively, and derive an explicit formula for the minimum order required to realize a positive entropy not exceeding a prescribed threshold.
发表机构
- Silesian University in Opava（奥帕瓦西里西亚大学）
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