{
  "abstract": "A classical problem in spectral graph theory is to determine, among all simple graphs with a fixed number of vertices and edges, those that maximize the number of spanning trees. Such graphs are called $t$-optimal. Petingi and Rodríguez [Discr. Math. 244 (2002), 351-373] reduced the construction of infinitely many $t$-optimal graphs to the determination of almost-regular graphs with a minimum number of induced $3$-paths, known as almost-regular $ν$-min graphs. They conjectured that the class of $d$-regular $ν$-min graphs is invariant under the addition or deletion of copies of the complete graph $K_{d+1}$, which would provide a method to construct large families of $t$-optimal graphs. This conjecture was recently verified for $d\\in \\{1,2,3,4,5\\}$ by Gong et al. [Discr. Math. 349(4), 2026]. In this work, we establish an asymptotic version of this conjecture. We prove that the class of $d$-regular $ν$-min graphs on at least $n$ vertices is invariant under the addition or deletion of copies of $K_{d+1}$ whenever $d\\geq 3$ and $n\\geq d^3$. In addition, we introduce a new method to construct $t$-optimal graphs based on the concept of trace-minimal graphs, originally introduced by Ábrego et al. [Linear Algebra Appl. 412 (2006), 161-221]. As a consequence, we obtain infinitely many graphs that are both trace-minimal and $t$-optimal.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36127",
  "author_total": 2,
  "authors": [
    {
      "name": "Pablo Romero"
    },
    {
      "name": "Louis Petingi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/av13awve4maka.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36127",
  "fulltext_url": null,
  "id": "av13awve4maka",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36127",
  "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": "W7224105736",
  "pdf_url": "https://arxiv.org/pdf/2610.36127",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/av13awve4maka.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36127",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36127"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Trace-minimal graphs maximizing the number of spanning trees",
  "title_text": "Trace-minimal graphs maximizing the number of spanning trees",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/av13awve4maka",
  "venue": "arXiv",
  "versions": []
}