{
  "abstract": "We study the singular Legendre operator on the edges of a finite connected graph, subject to renormalized continuity and Kirchhoff conditions at the vertices. An exact scattering trace formula and a reduction to the normalized adjacency matrix describe its spectrum. For the symmetrized spectral measure in r = λ + 1/4, we determine the singular support of its Fourier transform. Let L_G be the set of lengths kπ of nonempty closed edge walks with k steps (including immediate reversals). We prove the exact closed-walk length-spectrum identity sing supp µ (r)\n\nwhere g is the odd girth (the minimum length of an odd closed walk, with loops counted as one step), and g = ∞ for a bipartite graph. Every nonzero even period carries a Dirac mass and, except for the single-edge graph, a non-classical conormal principal-value term. At the surviving odd periods, the leading logarithmic conormal singularity is determined by g and the spectral invariant tr(A_G^g ) of the normalized adjacency matrix. The singular Legendre circle and equilateral stars illustrate these phenomena.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Elisha Falbel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3awq4d3z26fh7.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "3awq4d3z26fh7",
  "issns": [],
  "landing_url": "https://hal.science/hal-05805676v1",
  "language": "en",
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05805676/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3awq4d3z26fh7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:59Z",
      "native_id": "hal-05805676",
      "source": "hal",
      "url": "https://hal.science/hal-05805676v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "Singular Legendre operators on graphs: trace formulas and Fourier singular support",
  "title_text": "Singular Legendre operators on graphs: trace formulas and Fourier singular support",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:59Z",
  "url": "https://research.draconitic.com/paper/3awq4d3z26fh7",
  "venue": "HAL",
  "versions": []
}