{
  "abstract": "Let \\(G\\) be a graph with \\(m\\) edges and adjacency spectral radius\\(ρ(G)\\), and let \\(N(C_{2k+1},G)\\) denote the number of copies of \\(C_{2k+1}\\) in \\(G\\). For each fixed integer \\(k\\ge 2\\), define \\( g_k(m):=\\frac{k-1+\\sqrt{4m-k^2+1}}{2}. \\) Li, Zhai and Shu [European J. Combin., 2024] determined the spectral extremal threshold for odd cycles by proving that, for all sufficiently large \\(m\\), every \\(C_{2k+1}\\)-free graph \\(G\\) with \\(m\\) edges satisfies \\(ρ(G)\\le g_k(m)\\). We establish the asymptotically sharp supersaturation counterpart of their result. More precisely, for every fixed integer \\(k\\ge 2\\), we prove that \\[ \\inf_{\\substack{e(G)=m\\\\ ρ(G)>g_k(m)}} \\frac{N(C_{2k+1},G)}{m^k} = \\frac{\\lceil k^2/2\\rceil (k-1)!}{(k+1)^k}+o(1) \\qquad\\text{as }m\\to\\infty. \\] Thus every \\(m\\)-edge graph whose spectral radius exceeds the \\(C_{2k+1}\\)-free threshold contains at least \\[ \\Big( \\frac{\\lceil k^2/2\\rceil (k-1)!}{(k+1)^k}-o(1) \\Big)m^k \\] copies of \\(C_{2k+1}\\), and the leading constant is asymptotically best possible. In particular, taking \\(k=2\\), we obtain if \\( ρ(G)>\\frac{1+\\sqrt{4m-3}}{2}\\) then \\( N(C_5,G)\\ge \\left(\\frac{2}{9}-o(1)\\right)m^2, \\) with the constant \\(2/9\\) being asymptotically optimal. This answers a question of Chen, Li and Tang concerning the existence and the largest possible value of a constant \\(C>0\\) for which the same spectral condition guarantees at least \\(Cm^2\\) copies of \\(C_5\\). More generally, our result resolves a recent problem of Li, Lin, Liu and Zhang on spectral supersaturation for odd cycles. The proof combines spectral stability and resolvent analysis with estimates for odd spectral moments and a careful treatment of non-injective closed walks.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08077",
  "author_total": 3,
  "authors": [
    {
      "name": "Jiaqi Liu"
    },
    {
      "name": "Zhenzhen Lou"
    },
    {
      "name": "Shuang Sun"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d846qnzmeeqvz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08077",
  "fulltext_url": null,
  "id": "d846qnzmeeqvz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08077",
  "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",
      "math.CO"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7212001796",
  "pdf_url": "https://arxiv.org/pdf/2609.08077",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d846qnzmeeqvz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.08077",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08077"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Sharp edge-spectral supersaturation for odd cycles",
  "title_text": "Sharp edge-spectral supersaturation for odd cycles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/d846qnzmeeqvz",
  "venue": "arXiv",
  "versions": []
}