{
  "abstract": "In this paper, we study the higher-order ranking, by spanning-tree count, of graphs obtained from a complete graph by deleting a fixed number of edges. Using the edge-defect matrix determined by the deleted edges and the interaction number measuring the local overlap among them, we derive a stability inequality that quantitatively estimates the decrease in the number of spanning trees from the matching-deletion case. Combining this stability estimate with a classification of deletion graphs having small interaction number, we determine, up to isomorphism, the nine deletion graphs with the largest spanning-tree counts for $p\\geq6$ and $n\\geq2p$. We also clarify the relation between the interaction number, the local structure of the deletion graph, and the decrease in the number of spanning trees through a logarithmic expansion of the normalized spanning-tree count.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33979",
  "author_total": 2,
  "authors": [
    {
      "name": "Shunya Tamura"
    },
    {
      "name": "José Luis Palacios"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1xg5pxj66ykqb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33979",
  "fulltext_url": null,
  "id": "1xg5pxj66ykqb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33979",
  "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": "W7214790729",
  "pdf_url": "https://arxiv.org/pdf/2609.33979",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1xg5pxj66ykqb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33979",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33979"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Edge-Defect Spectral Methods for Higher-Order Rankings of Spanning Tree Counts",
  "title_text": "Edge-Defect Spectral Methods for Higher-Order Rankings of Spanning Tree Counts",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/1xg5pxj66ykqb",
  "venue": "arXiv",
  "versions": []
}