{
  "abstract": "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17334",
  "author_total": 1,
  "authors": [
    {
      "name": "Rostislav Klech"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3x007425ebagf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17334",
  "fulltext_url": null,
  "id": "3x007425ebagf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17334",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7213392198",
  "pdf_url": "https://arxiv.org/pdf/2609.17334",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3x007425ebagf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.17334",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17334"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Generating Functions and the Entropy Hierarchy of Strongly Connected Digraphs",
  "title_text": "Generating Functions and the Entropy Hierarchy of Strongly Connected Digraphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/3x007425ebagf",
  "venue": "arXiv",
  "versions": []
}