{
  "abstract": "For a graph $G$ of order $n$, put $σ(G)=(λ_1(G)+λ_n(G))/n$. We determine the first three largest values of this invariant among nonbipartite distance-regular graphs of diameter three and odd girth at least seven. The unique maximizer is the folded $7$-cube, with value $1/32$; the unique second maximizer is the Odd graph $O_4$, with value $1/35$; and the unique third maximizer is $C_7$, with value $2(1-\\cos(π/7))/7$. More precisely, every other graph in the class satisfies $σ(G)<1/36$. This answers Problem~11 of Abiad, Taranchuk and van Veluw in \\emph{Electronic Journal of Combinatorics} 33(2) (2026), P2.31. The proof combines established local multiplicity and odd-moment bounds: the condition $σ(G)\\geq1/36$ forces the valency to be at most $182$. An exhaustive certificate using only integer and rational arithmetic then leaves three intersection arrays. The complete certificate is publicly available, and neither a classification of generalized odd graphs nor the $Q$-polynomial property is assumed. The odd-girth theorem gives the same extremal conclusions for connected $\\{C_3,C_5\\}$-free graphs with at most four distinct adjacency eigenvalues, without assuming regularity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11729",
  "author_total": 1,
  "authors": [
    {
      "name": "Qi Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e0g2t1red87f2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11729",
  "fulltext_url": null,
  "id": "e0g2t1red87f2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11729",
  "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": "W7212268356",
  "pdf_url": "https://arxiv.org/pdf/2609.11729",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e0g2t1red87f2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11729",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11729"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Spectral bipartiteness in generalized odd graphs of diameter three",
  "title_text": "Spectral bipartiteness in generalized odd graphs of diameter three",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/e0g2t1red87f2",
  "venue": "arXiv",
  "versions": []
}