{
  "abstract": "The generalized pancake graph $P(m,n)$ is the Cayley graph of the group of colored permutations $\\mathbb{Z}_m\\wr S_n=(\\mathbb{Z}_m)^n\\rtimes S_n$ generated by generalized prefix reversals. In this paper, we establish that, for all $m,n\\geq2$, the spectral gap $γ(P(m,n))$ of the normalized Laplacian satisfies $α_m/n\\leqγ(P(m,n))\\leq1/n$, where $α_m$ is a positive constant that depends only on $m$. As a consequence, for every fixed $m\\geq2$, $γ(P(m,n))$ is $Θ_m(1/n)$ as $n\\to\\infty$. The proof combines Cesi's semi-recursive spectral-gap inequality with a Fourier decomposition of the appropriate operators associated with a coset Schreier graph of color-position pairs. For fixed $n\\geq2$, we also establish that $γ(P(m,n))$ is $Θ_n(m^{-2})$ as $m\\to\\infty$. This disproves a conjecture of Blanco and Buehrle asserting that, for fixed $n$, the corresponding undirected generalized pancake graphs form an expander family. Additionally, we present a counterexample to a recent conjecture of Greaves and Zhu concerning equality between the spectral gaps of the full Cayley graph and the associated coset Schreier graph.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.15398",
  "author_total": 1,
  "authors": [
    {
      "name": "Saúl A. Blanco"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5hrz3m0zsahhe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.15398",
  "fulltext_url": null,
  "id": "5hrz3m0zsahhe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.15398",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7203673897",
  "pdf_url": "https://arxiv.org/pdf/2608.15398",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5hrz3m0zsahhe.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2608.15398",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.15398"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On the Laplacian spectral gap of generalized pancake graphs",
  "title_text": "On the Laplacian spectral gap of generalized pancake graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/5hrz3m0zsahhe",
  "venue": "arXiv",
  "versions": []
}