{
  "abstract": "We characterize adjacency perfect state transfer in connected unicyclic graphs: the transfer vertices are opposite degree-two vertices of a four-cycle, and the two attached trees satisfy a characteristic-polynomial condition. Except for the four-cycle, the transfer pair is unique and the minimum transfer time is $π/\\sqrt2$. Outside the four-cycle, we obtain sharp lower bounds of $46$ on the order and $10$ on the maximum degree, and classify all examples of maximum degree at most $20$. A finite inverse construction enumerates all realizations of a prescribed eigenvalue support at the transfer vertices. For supports of size at least seven, we establish the sharp maximum-degree lower bound of $21$. Every seven-element support $\\{0,\\pm\\sqrt2,\\pm u\\sqrt2,\\pm v\\sqrt2\\}$, with odd integers $3\\leq u<v$, is realizable, and every such realization has attachment depth at most four. Explicit branching rules give a minimum-order recurrence. Among all connected unicyclic graphs admitting perfect state transfer with seven-element eigenvalue support, the minimum order is $218$, attained uniquely up to isomorphism.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41572",
  "author_total": 1,
  "authors": [
    {
      "name": "Xingkun Song"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aeaa9hp78feht.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41572",
  "fulltext_url": null,
  "id": "aeaa9hp78feht",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41572",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41572",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aeaa9hp78feht.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41572",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41572"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Perfect state transfer on unicyclic graphs",
  "title_text": "Perfect state transfer on unicyclic graphs",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/aeaa9hp78feht",
  "venue": "arXiv",
  "versions": []
}