{
  "abstract": "For a finite group $G$ and a symmetric connection set $S$, we study the distance Laplacian ($D^L$) and distance signless Laplacian ($D^Q$) spectrum of the Cayley graph $\\Cay(G,S)$ for $G$ a finite nilpotent group. Using the internal direct product decomposition over Sylow subgroups, we reduce every question to a single (possibly non-abelian) prime-power factor. In this setting, the required tools are furnished by ordinary representation theory rather than character theory. Based on this framework, we then derive a complete and explicit formula for every eigenvalue, with multiplicity, of the distance matrix, distance Laplacian, and distance signless Laplacian of $\\Cay(G,S)$ for $G$ nilpotent: each eigenvalue is indexed by a tuple of irreducible characters of the Sylow factors. Finally, we use the explicit spectrum, together with Perron--Frobenius theory, to determine the eigenvalue-free interval of $D^L$ and $D^Q$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13429",
  "author_total": 2,
  "authors": [
    {
      "name": "Pallabi Manna"
    },
    {
      "name": "Santanu Mandal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/542hx25wz3snz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13429",
  "fulltext_url": null,
  "id": "542hx25wz3snz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13429",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7221568774",
  "pdf_url": "https://arxiv.org/pdf/2610.13429",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/542hx25wz3snz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13429",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13429"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "$D^L$, $D^Q$ Spectrum and Eigenvalue-free Interval of Cayley Graphs over Finite Nilpotent Groups",
  "title_text": "D^L, D^Q Spectrum and Eigenvalue-free Interval of Cayley Graphs over Finite Nilpotent Groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/542hx25wz3snz",
  "venue": "arXiv",
  "versions": []
}