{
  "abstract": "The problem of maximizing vertex-degree-based topological indices over trees with a prescribed order and maximum degree $Δ$ has traditionally relied on ad-hoc combinatorial arguments. In this paper, we establish a unified linear programming approach that systematically resolves these extremal problems. By formulating the optimization over degree-pair multiplicities, the search for an extremal tree reduces to minimizing a discrete dual penalty. Treating vertex degrees as continuous variables yields an analytical single-variable test condition that guarantees global optimality. We apply this framework across a diverse collection of invariants. We prove that the standard basic feasible solution is optimal for the Tangent Sombor, $SO_1$, $SO_2$, Albertson, and $σ$-irregularity indices for all $Δ\\ge 3$. However, the dual optimality conditions establish exact analytical thresholds at which the standard basis loses optimality, necessitating a transition to alternative basic feasible solutions. For the Arithmetic-Geometric ($AG$) index, the standard basis is optimal strictly for $Δ\\ge 9$; for smaller maximum degrees ($3 \\le Δ\\le 8$), optimization shifts to an alternative edge basis, allowing us to explicitly construct and characterize all extremal trees across congruence classes, including chemical trees ($Δ\\in \\{3, 4\\}$). Conversely, for the Triangle Volume index ($SO_{11}$), the standard basis is optimal exclusively for $Δ\\le 8$, while for $Δ\\ge 9$ we analytically derive a novel, degree-dependent optimal basis to explicitly construct its extremal configurations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20364",
  "author_total": 1,
  "authors": [
    {
      "name": "Milan Bašić"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9bgtkxyf3n78m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20364",
  "fulltext_url": null,
  "id": "9bgtkxyf3n78m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20364",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.20364",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9bgtkxyf3n78m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.20364",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20364"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Unified Linear Programming Approach for Maximizing Vertex-Degree-Based Indices of Trees",
  "title_text": "A Unified Linear Programming Approach for Maximizing Vertex-Degree-Based Indices of Trees",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/9bgtkxyf3n78m",
  "venue": "arXiv",
  "versions": []
}