{
  "abstract": "We study the maximum order $v(k,θ)$ of a connected $k$-regular graph whose second largest eigenvalue is at most $θ$. For every real $θ\\geq1$, we prove that \\[ T(θ) =\\limsup_{k\\to\\infty}\\bigl(v(k,θ)-2k\\bigr) =2\\left\\lfloor1+\\frac{θ^2}{4}\\right\\rfloor. \\] In particular, $T(θ)>2θ$ for every real $θ\\geq4$, disproving the Yang-Koolen conjecture. We also determine $v(k,θ)$ for all sufficiently large $k$ on explicit intervals of $θ$. We prove a jump in the second largest eigenvalue above $1$. For each fixed $1\\leqθ<2$ and all sufficiently large $k$, every connected $k$-regular graph of order at least $2k+2$ whose second largest eigenvalue is at most $θ$ is the crown graph. For $θ=\\sqrt2$, we prove that $v(k,\\sqrt2)=2k+2$ for every $k\\geq13$. Finally, we obtain two new upper bounds on $v(k,θ)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00591",
  "author_total": 1,
  "authors": [
    {
      "name": "Vishal Gupta"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8feqyhm0ypwsc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00591",
  "fulltext_url": null,
  "id": "8feqyhm0ypwsc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00591",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00591",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8feqyhm0ypwsc.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00591",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00591"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A spectral Moore problem and the Yang-Koolen conjecture",
  "title_text": "A spectral Moore problem and the Yang-Koolen conjecture",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/8feqyhm0ypwsc",
  "venue": "arXiv",
  "versions": []
}