{
  "abstract": "For a connected graph $G$, let $τ(G)$ denote the maximum number of pairwise edge-disjoint spanning trees, and let $ρ(G)$ be its adjacency spectral radius. For integers $n\\ge1$, $k\\ge2$, and $k\\leδ\\le2k-1$, let $\\mathcal{G}_{n,δ}$ be the class of connected $n$-vertex graphs with minimum degree $δ$, and let $\\mathcal{L}_{\\mathcal{H}}^{2}(n,k,δ)\\subseteq\\mathcal{G}_{n,δ}$ be the exceptional family introduced by Chang, Li, and Zhang. For sufficiently large $n$, the sharp adjacency-spectral threshold for $τ(G)\\ge k$ is determined by maximizing $ρ(G)$ over this family. Set $h=δ-k$. For each fixed admissible pair $(k,δ)$ and all sufficiently large $n$, we determine the maximum and minimum adjacency spectral radii in $\\mathcal{L}_{\\mathcal{H}}^{2}(n,k,δ)$. In the core--placement representation, let $M$ be the missing-edge graph of the bounded core. For $h\\ge2$, the unique maximizer, up to isomorphism, satisfies $M\\cong K_{1,h}\\cup(h+1)K_1$, with the exceptional edges nested on the large-clique side. Hence the matching configuration proposed in Conjecture~2 of Chang--Li--Zhang is not extremal. For every $h\\ge1$, the unique minimizer satisfies $M\\cong hK_2\\cup2K_1$, with the $2h$ exceptional edges having distinct endpoints on the large-clique side. We also settle the cases $h=0,1$ and all equality cases. The proof combines an exact core--placement parametrization with a uniform Schur-complement resolvent expansion. The first candidate-dependent coefficient is affine in $\\sum_{x\\in V(M)}d_M(x)^2$, while the first placement-sensitive coefficient is a squared-load functional. Equitable quotient matrices yield the two extremal radii, and the maximizing graph gives the sharp global adjacency-spectral threshold.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19655",
  "author_total": 2,
  "authors": [
    {
      "name": "Xuanyu Cao"
    },
    {
      "name": "Chunxiang Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ap4qq38yexyv0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19655",
  "fulltext_url": null,
  "id": "ap4qq38yexyv0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19655",
  "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": "W7213596869",
  "pdf_url": "https://arxiv.org/pdf/2609.19655",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ap4qq38yexyv0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.19655",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19655"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Maximum and Minimum Spectral Radii in an Exceptional Family for Edge-Disjoint Spanning Trees",
  "title_text": "Maximum and Minimum Spectral Radii in an Exceptional Family for Edge-Disjoint Spanning Trees",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/ap4qq38yexyv0",
  "venue": "arXiv",
  "versions": []
}