{
  "abstract": "For the circulant graph $C_n(1,2)$ with $n\\ge10$ even, the $\\F_2$ system requiring every quadrilateral to be unbalanced is consistent and its solutions form exactly four switching classes. We show that the class containing the signing which is $+1$ on step-$1$ edges and $(-1)^i$ on step-$2$ edges has spectrum $\\{\\pm2\\sqrt{\\cos^2θ_k+\\cos^2 2θ_k}\\}$ and spectral radius exactly $2\\sqrt2$, well below the Kesten bound $2\\sqrt3$; that the quadrilateral system is equivalent to alternating triangle fluxes, so that the four classes are coordinatized by $(τ_0,α)$ and the spectral radius depends only on the Hamilton-cycle holonomy $α$; and that the two twisted classes attain $ρ_-(n)=2\\sqrt{\\cos^2(π/n)+\\cos^2(2π/n)}<2\\sqrt2$. Exhaustive enumeration of all $2^{n+1}$ switching classes for $n\\in\\{8,10,12,14,16,18\\}$ shows that $ρ_-(n)$ is the global minimum in every case, and we conjecture this for all even $n$; the lower bound is a flux-minimization statement in the sense of Lieb's flux-phase theorem. For odd $n$ the quadrilateral system is inconsistent.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.18334",
  "author_total": 1,
  "authors": [
    {
      "name": "Vaibhav Suvagiya"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/003k3cfkqh2kv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.18334",
  "fulltext_url": null,
  "id": "003k3cfkqh2kv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.18334",
  "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": "W7170094744",
  "pdf_url": "https://arxiv.org/pdf/2607.18334",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/003k3cfkqh2kv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2607.18334",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.18334"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Signed circulants at the Ramanujan bound",
  "title_text": "Signed circulants at the Ramanujan bound",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/003k3cfkqh2kv",
  "venue": "arXiv",
  "versions": []
}