{
  "abstract": "Let $Y_{n,r,q}$ be the graph obtained from the Turán graph $T_{n,r}$ by adding $q$ pairwise disjoint edges inside a largest part, and let $c(n,F)$ be the minimum number of copies of $F$ created by adding a single edge to $T_{n,r}$. Fang, Li, Lin and Ma proved that for every color-critical graph $F$ with $χ(F)=r+1$, there exists a constant $δ_F>0$ such that for all sufficiently large $n$ and all $1\\le q\\le δ_F \\sqrt{n}$, the condition $λ(G)\\geλ(Y_{n,r,q})$ forces at least $q\\, c(n,F)$ copies of $F$. The bound $q=O(\\sqrt{n}\\,)$ is tight up to a constant factor, in contrast to the linear order $n$ of the edge setting of Mubayi, Pikhurko and Yilma, but the exact constant $δ_F$ remained unknown for any $F$. In this paper, building on a structural result of Fang, Li, Lin and Ma, we determine the threshold $δ_F$ when $F$ is a clique and an odd cycle. For every $r\\ge2$, we denote $δ_r :=(1-\\tfrac1r)\\sqrt2$ and prove that for every $\\varepsilon>0$, if $n$ is sufficiently large and $1\\le q\\le(δ_r-\\varepsilon)\\sqrt n$, then every $n$-vertex graph $G$ with $λ(G)\\geλ(Y_{n,r,q})$ contains at least $q\\,c(n,K_{r+1})$ copies of $K_{r+1}$, and $δ_r$ is best possible. For odd cycles, the threshold is $1/\\sqrt2$. For every $k\\ge1$ and $\\varepsilon>0$, if $n$ is sufficiently large and $1\\le q\\le(1/\\sqrt2-\\varepsilon)\\sqrt n$, then every $n$-vertex graph $G$ with $λ(G)\\geλ(Y_{n,2,q})$ contains at least $q\\,c(n,C_{2k+1})$ copies of $C_{2k+1}$, and $1/\\sqrt2$ is best possible. Our results determine both the exact count of copies and the optimal range of $q$. The behavior in the spectral setting differs from the classical edge setting, in which the range of $q$ is of order $n$ and the threshold is $1/r$ for cliques by Lovász and Simonovits, and $1/2$ for odd cycles by Pikhurko and Yilma.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06038",
  "author_total": 2,
  "authors": [
    {
      "name": "Hongzhang Chen"
    },
    {
      "name": "Yongtao Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/59cg8y56qkzfg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06038",
  "fulltext_url": null,
  "id": "59cg8y56qkzfg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06038",
  "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"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7220691385",
  "pdf_url": "https://arxiv.org/pdf/2610.06038",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/59cg8y56qkzfg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.06038",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06038"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Optimal spectral supersaturation for cliques and odd cycles",
  "title_text": "Optimal spectral supersaturation for cliques and odd cycles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/59cg8y56qkzfg",
  "venue": "arXiv",
  "versions": []
}