{
  "abstract": "We introduce a weighted adjacency-degree matrix \\(A_{fg}(G)\\) with edge weights \\(f(d_i,d_j)\\) and diagonal entries \\(g(d_i)\\), unifying adjacency, Laplacian, signless Laplacian, \\(A_α\\), ABC, Randić, Sombor and related matrices. For dendrimer trees \\(D_{n,k}\\) and Bethe trees \\(B_{n,k}\\), the characteristic polynomial is factorized through one recursive sequence \\(P_{fg,n}\\). When \\(f^2(1,k)=f^2(k,k)\\), this sequence admits a Chebyshev reduction to \\( U_{j+1}(x)+δU_j(x)=0, \\) covering \\(L/L^+\\) and \\(A_α\\); explicit cosine spectra occur only in special cases, such as the adjacency matrix and the \\(L/L^+\\) end factor of \\(B_{n,k}\\). We derive positive-semidefinite energy formulas, Gershgorin criteria, spectral-gap estimates, interlacing and non-interlacing results, partial eigenvalue-coincidence information, and McClelland- and Koolen--Moulton-type bounds. Several known results are recovered as special cases of this unified framework.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10107",
  "author_total": 1,
  "authors": [
    {
      "name": "Metrose Metsidik"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1r0t38z9zwaev.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10107",
  "fulltext_url": null,
  "id": "1r0t38z9zwaev",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10107",
  "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",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7221017206",
  "pdf_url": "https://arxiv.org/pdf/2610.10107",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1r0t38z9zwaev.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.10107",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10107"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Unified Spectral Framework for Weighted Adjacency-Degree Matrices of Dendrimer and Bethe Trees",
  "title_text": "A Unified Spectral Framework for Weighted Adjacency-Degree Matrices of Dendrimer and Bethe Trees",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/1r0t38z9zwaev",
  "venue": "arXiv",
  "versions": []
}