{
  "abstract": "We develop an affine $\\mathbb F_2$ framework for structured signings of regular graphs. A family-averaging identity converts even spectral moments into parity-weighted closed-walk counts supported on the span of prescribed short even cycles, while a kernel-averaged Ihara identity gives the corresponding decomposition at the non-backtracking level. We give a finite-scale bounded-rank counting estimate and a conditioning corollary showing that, on bicycle-free graph sequences, any parity family of uniformly bounded codimension contains near-Ramanujan signings whenever the corresponding random-signing theorem applies. The latter is a transfer statement rather than a new concentration theorem. Finally, on $C_n(1,2)$ for even $n\\ge10$, the quadrilateral-unbalanced family has exactly four switching classes and its twisted classes attain $ρ_-(n)=2\\sqrt{\\cos^2(π/n)+\\cos^2(2π/n)}$; a period-$8$ signing has spectral radius $r_*=2.793604493334841\\ldots$ for every positive multiple of $8$. Thus for $n=8m\\ge32$ the constrained minimum is strictly larger than a value attained by an unrestricted signing, while equality of $r_*$ with the unrestricted minimum remains conjectural.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.17343",
  "author_total": 1,
  "authors": [
    {
      "name": "Vaibhav Suvagiya"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a6g0v8kh23v7b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.17343",
  "fulltext_url": null,
  "id": "a6g0v8kh23v7b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.17343",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7169879885",
  "pdf_url": "https://arxiv.org/pdf/2607.17343",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a6g0v8kh23v7b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2607.17343",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.17343"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Parity families and signed spectra: kernel averaging, near-Ramanujan bounds, and exact circulant models",
  "title_text": "Parity families and signed spectra: kernel averaging, near-Ramanujan bounds, and exact circulant models",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/a6g0v8kh23v7b",
  "venue": "arXiv",
  "versions": []
}