{
  "abstract": "We present a general algebraic method for maximizing a vertex-degree-based (VDB) topological index $TI(G) = \\sum_{uv \\in E(G)} f(d(u),d(v))$ over the class of $n$-vertex trees with a prescribed maximum degree $Δ\\ge 3$. By expressing the edge multiplicities of the basis $\\mathcal{B} = \\{(1,Δ), (Δ,Δ)\\}$ via degree-sum identities, we reduce the maximization of $TI(T)$ to the minimization of an explicit discrete penalty function $G(x,y)$ over the non-basic edges. We formulate five analytical conditions on the weight function $f(x,y)$ that dictate the optimal edge distributions, yielding a complete characterization of the extremal trees across all congruence classes of $n \\pmod{Δ- 1}$. Additionally, we provide their explicit construction by attaching leaves to a core of internal vertices. Applying this framework, we obtain the complete characterization of extremal $Δ$-trees for the Degree-Ratio Sombor ($DRSO$), Symmetric Division Deg ($SDD$), and Hyperbolic Sombor ($HSO$) indices for all $Δ\\ge 3$. Furthermore, the analytical conditions identify the exact degree thresholds for the Arithmetic-Geometric ($AG$) and Diminished Sombor ($DSO$) indices, showing that the same extremal family maximizes $AG$ if and only if $3 \\le Δ\\le 8$, and $DSO$ if and only if $3 \\le Δ\\le 4$. Finally, since changing the edge count from $m = n - 1$ to $m = n - 1 + k$ affects only the constant term in the decomposition while leaving the penalty function $G(x,y)$ invariant, the method extends directly to connected $k$-cyclic graphs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34361",
  "author_total": 1,
  "authors": [
    {
      "name": "Milan Bašić"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fn07jtww3sj27.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34361",
  "fulltext_url": null,
  "id": "fn07jtww3sj27",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34361",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7223796245",
  "pdf_url": "https://arxiv.org/pdf/2610.34361",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fn07jtww3sj27.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34361",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34361"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A general algebraic characterization of extremal $Δ$-trees and $k$-cyclic graphs for degree-homogeneous VDB indices",
  "title_text": "A general algebraic characterization of extremal Δ-trees and k-cyclic graphs for degree-homogeneous VDB indices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/fn07jtww3sj27",
  "venue": "arXiv",
  "versions": []
}