{
  "abstract": "Let $G$ be a finite simple graph on the vertex set $[n]$, and let $Γ(G)$ be the Cayley graph of the symmetric group $S_n$ generated by the transpositions corresponding to the edges of $G$. We study the inverse spectral problem for the assignment $G\\mapsto Γ(G)$ through the group-algebra operator $A_G=\\sum_{\\{i,j\\}\\in E(G)}(ij)$ and its regular spectral moments $M_k(G)=\\operatorname{tr}(A_G^k)$. The moments admit two all-order forms: a universal support expansion in non-induced subgraph counts of $G$, and a representation-theoretic expansion over the irreducible representations of $S_n$. We compute the first nontrivial moments through order six. In particular, the fourth moment already contains the Coxeter triangle relation and determines the signed local invariant $6N_{K_3}(G)-N_{P_3}(G)$, rather than $N_{P_3}(G)$ alone. Consequently, for triangle-free graphs the spectrum determines the second moment of the degree sequence. We also give a simplified sixth-moment formula for triangle-free graphs, showing that in this class the spectrum determines $5N_{K_{1,3}}(G)+2N_{P_4}(G)+12N_{C_4}(G)$. Further structural consequences include bipartiteness and spectral symmetry of $Γ(G)$, a precise connectedness test, and the identification of the standard irreducible component with the shifted graph Laplacian $mI-L_G$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14373",
  "author_total": 1,
  "authors": [
    {
      "name": "Boris Shapiro"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9821m0gkds9nb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14373",
  "fulltext_url": null,
  "id": "9821m0gkds9nb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14373",
  "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.CO"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7221757523",
  "pdf_url": "https://arxiv.org/pdf/2610.14373",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9821m0gkds9nb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14373",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14373"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Transposition Cayley Spectra and Subgraph Counts",
  "title_text": "Transposition Cayley Spectra and Subgraph Counts",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/9821m0gkds9nb",
  "venue": "arXiv",
  "versions": []
}