{
  "abstract": "Let $\\mathcal S(T)$ be the adjacency spectral center of a tree $T$, and let $\\mathcal C(T)$ be its characteristic set. We determine the largest possible separation $d(T) := \\operatorname{dist}_T(\\mathcal{S}(T), \\mathcal{C}(T))$ among trees of every order $n\\ge3$. Writing $Δ_n:=\\max_{|V(T)|=n}d(T)$, we prove $Δ_n=0\\quad(3\\le n\\le11),\\quad Δ_{12}=1$, and $Δ_n=\\left\\lfloor\\frac{n-11}{2}\\right\\rfloor \\quad(n\\ge13)$. The argument rests on a simple opposition between two rooted-tree weights. An endpoint-rooted path minimizes adjacency spectral radius, but maximizes bottleneck Perron value. A one-sided replacement by a path therefore cannot decrease the distance between the two centers. Quantitatively, this gives the sharp estimate $|V(T)|\\ge 2d(T)+11 \\quad(d(T)\\ge2)$. A preliminary six-vertex barrier shows that disjoint center sets require at least twelve vertices, and the four-leaf broom is extremal for every $n\\ge12$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16943",
  "author_total": 4,
  "authors": [
    {
      "name": "Chaochao Zhu"
    },
    {
      "name": "Shipei Hu"
    },
    {
      "name": "Jingfu Huang"
    },
    {
      "name": "Qin Yue"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bmrrgtgbnd24z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16943",
  "fulltext_url": null,
  "id": "bmrrgtgbnd24z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16943",
  "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": "W7213391292",
  "pdf_url": "https://arxiv.org/pdf/2609.16943",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bmrrgtgbnd24z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16943",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16943"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Distance Between the Adjacency Spectral Center and the Characteristic Set of a Tree",
  "title_text": "The Distance Between the Adjacency Spectral Center and the Characteristic Set of a Tree",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/bmrrgtgbnd24z",
  "venue": "arXiv",
  "versions": []
}