{
  "abstract": "Let $Γ$ be a distance-regular graph with valency $k$, diameter $D\\ge3$, and eigenvalues $k=θ_0>θ_1>\\cdots>θ_D$. Put $b\\coloneqq b_1/(θ_1+1)$, and let $1=u_0>u_1>\\cdots>u_D$ be the standard sequence associated with $θ_1$. Lv and Koolen asked whether the negative of the smallest eigenvalue of a distance-regular graph can be bounded above in terms of its diameter $D$ and the parameter $b$ alone. In this paper, we answer this question affirmatively by proving that $-θ_D\\le\\prod_{i=2}^D(1-u_i)/(u_1-u_i) <(b+1)^{D-1}$. Equality in the product bound holds if and only if $Γ$ is $Q$-polynomial with respect to the primitive idempotent corresponding to $θ_1$. For every integer $D\\ge4$, we further establish the explicit bound $-θ_D\\le b^{D-1}+b^{D-2} +\\frac{27}{2}(D-2)b^{D-3}$ whenever $b\\ge\\max\\{8,\\sqrt{12(D-2)}\\}$. Both leading coefficients are best possible for each fixed $D\\ge4$. The proof uses a spectral product theorem of Micchelli and Willoughby applied to Schur multiplication in the Bose--Mesner algebra.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37010",
  "author_total": 3,
  "authors": [
    {
      "name": "Zhen-Mu Hong"
    },
    {
      "name": "Fu-Tao Hu"
    },
    {
      "name": "Chenhui Lv"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e600am9afr0xp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37010",
  "fulltext_url": null,
  "id": "e600am9afr0xp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37010",
  "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": "W7224114182",
  "pdf_url": "https://arxiv.org/pdf/2610.37010",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e600am9afr0xp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.37010",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37010"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Resolving a Problem of Lv and Koolen on the Smallest Eigenvalue of Distance-Regular Graphs",
  "title_text": "Resolving a Problem of Lv and Koolen on the Smallest Eigenvalue of Distance-Regular Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/e600am9afr0xp",
  "venue": "arXiv",
  "versions": []
}