{
  "abstract": "For every fixed $r\\ge3$ and all sufficiently large $n$, we determine the largest adjacency spectral radius of an $n$-vertex graph with fewer than $q c_r(n)$ copies of $K_{r+1}$, where $1\\le q<n/r$. Here $c_r(n)$ is the number of copies created by adding one edge to a largest part of the Turán graph. We also determine all extremal graphs. In most cases the extremal graph is obtained from an almost balanced complete multipartite graph by adding a star in one part. Two small values of $q$ require separate constructions, and an additional transition occurs when $r=3$ and $n\\equiv2\\pmod3$. Under the non-strict constraint, the unique extremal graph is obtained by adding a $q$-edge star to a largest part of the Turán graph. We determine the difference between the strict and non-strict values and prove that the sharp matching threshold is $\\sqrt2(r-1)/r$. The proof treats separately the ranges $q=o(n)$, $q/m\\toγ\\in(0,1)$, and $q/m\\to1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03337",
  "author_total": 2,
  "authors": [
    {
      "name": "Qi Wu"
    },
    {
      "name": "Yong Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d748vfjbyvmps.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03337",
  "fulltext_url": null,
  "id": "d748vfjbyvmps",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03337",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7220469054",
  "pdf_url": "https://arxiv.org/pdf/2610.03337",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d748vfjbyvmps.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.03337",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03337"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Strict spectral supersaturation for cliques: extremal graphs and sharp thresholds",
  "title_text": "Strict spectral supersaturation for cliques: extremal graphs and sharp thresholds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/d748vfjbyvmps",
  "venue": "arXiv",
  "versions": []
}