{
  "abstract": "We introduce and study equivariant analytic spectral invariants associated with the Açıkmeşe lift of a graph with self-loops $G_S$. The canonical $\\mathbb{Z}_2$-action yields an orthogonal isotypic decomposition of the lifted Laplacian into the anti-symmetric part $\\mathcal{L}(G_S)$ and the symmetric part $M_{sym}$, from which we prove that the spectrum of $\\mathcal{L}(G_S)$ is exactly the even-indexed spectrum of the lifted Laplacian. We express twisted moments of these blocks as traces of the generalised twisted moment operator against the corresponding $\\mathbb{Z}_2$-projections, and obtain a tight upper bound under certain restriction with a characterization of the equality case. We introduce the equivariant heat character and establish a trace-norm stability estimate. For the matrix element of the resolvent $(pI+\\mathcal{L}(G_S))^{-1}$ associated with the loop vector, we derive a Laplace-transform identity for the equivariant heat character, a determinant formula, a spectral representation, and a Laurent expansion. We further characterize the case where $S$ is a union of connected components through resolvent and equivariant heat-character identities, and obtain explicit equivariant heat characters for joins of full-loop graphs and the line graph of a full-loop connected regular graph. Finally, we introduce the regularized equivariant heat integral and derive spectral and trace formulas for it.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29982",
  "author_total": 1,
  "authors": [
    {
      "name": "Johnny Lim"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fybg0s8a4r30m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29982",
  "fulltext_url": null,
  "id": "fybg0s8a4r30m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29982",
  "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.SP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7204929136",
  "pdf_url": "https://arxiv.org/pdf/2608.29982",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fybg0s8a4r30m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29982",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29982"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Equivariant Analytic Spectral Invariants of Açıkmeşe Lifts of Graphs with Self-Loops",
  "title_text": "Equivariant Analytic Spectral Invariants of Açıkmeşe Lifts of Graphs with Self-Loops",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/fybg0s8a4r30m",
  "venue": "arXiv",
  "versions": []
}