{
  "abstract": "Let $Γ$ be a distance-regular graph with diameter $D \\geq 3$, and let $θ$ be a nonzero eigenvalue of $Γ$ distinct from its valency. Motivated by the geometric parameter sequences of geometric distance-regular graphs, we introduce analogous sequences defined recursively from the intersection numbers of $Γ$ and $θ$, without assuming a geometric structure. We show that the recursion is well defined if and only if all terms of the standard sequence corresponding to $θ$ are nonzero and consecutive terms are distinct. In particular, the geometric parameter sequences associated with the smallest eigenvalue always exist. Suppose that $Γ$ is $Q$-polynomial with respect to an eigenvalue ordering $θ_0,θ_1, \\ldots,θ_D$. Let $\\{φ_i\\}_{i=1}^{D}$ and $\\{ϕ_i\\}_{i=1}^{D}$ be the first and second split sequences of the primary Leonard system, respectively, and let $β$ denote its fundamental parameter. We prove that the geometric parameter sequences associated with $θ_D$ exist if and only if $φ_i\\neqϕ_i$ for every $1\\leq i\\leq D$, and if and only if $β\\neq-2$ or $D$ is even. When the sequences exist, we obtain closed formulas in terms of the primary parameter array. We introduce the split ratio sequence $ρ_i=φ_i/ϕ_i$. Using this sequence, we obtain a bidiagonal factorization of $B-θ_D I$, where $B$ is the intersection matrix of $Γ$. This factorization remains valid even when the geometric parameter sequences associated with $θ_D$ do not exist. If $Γ$ is geometric and $θ_D$ is its smallest eigenvalue, then the geometric parameter sequences associated with $θ_D$ coincide with those of $Γ$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26522",
  "author_total": 3,
  "authors": [
    {
      "name": "Jae-Ho Lee"
    },
    {
      "name": "Jongyook Park"
    },
    {
      "name": "Ian Seong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dzygq7e5v86dx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26522",
  "fulltext_url": null,
  "id": "dzygq7e5v86dx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26522",
  "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",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7223037361",
  "pdf_url": "https://arxiv.org/pdf/2610.26522",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dzygq7e5v86dx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.26522",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26522"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Geometric parameter sequences for $Q$-polynomial distance-regular graphs",
  "title_text": "Geometric parameter sequences for Q-polynomial distance-regular graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/dzygq7e5v86dx",
  "venue": "arXiv",
  "versions": []
}