{
  "abstract": "Let $k\\ge 3$ and let $n=km$. A $K_k$-factor in an $n$-vertex graph is a collection of $m$ vertex-disjoint copies of $K_k$ that covers the entire vertex set. We determine the maximum adjacency spectral radius of an $n$-vertex graph containing no $K_k$-factor when $m\\ge 2k-1$. More precisely, we prove that every such graph $G$ satisfies \\[ ρ(G)\\le ρ(H_{n,k}), \\qquad H_{n,k}=K_{k-2}\\vee\\bigl(K_{n-k+1}\\cup K_1\\bigr), \\] with equality if and only if $G\\cong H_{n,k}$. Equivalently, the unique extremal graph is obtained from $K_{n-1}$ by adding one vertex adjacent to exactly $k-2$ vertices of the clique. Our proof combines a decomposition lemma for sparse complements, derived from the Hajnal--Szemerédi theorem, with the Motzkin--Straus inequality and spectral estimates based on quotient matrices and the Rayleigh quotient.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21529",
  "author_total": 3,
  "authors": [
    {
      "name": "Cunxiang Duan"
    },
    {
      "name": "Tingting Han"
    },
    {
      "name": "Lin-Peng Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a3wv6ht2qwhhb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21529",
  "fulltext_url": null,
  "id": "a3wv6ht2qwhhb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21529",
  "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",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7213952452",
  "pdf_url": "https://arxiv.org/pdf/2609.21529",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a3wv6ht2qwhhb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.21529",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21529"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Spectral Extremal Graphs without a $K_k$-Factor",
  "title_text": "Spectral Extremal Graphs without a K_k-Factor",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/a3wv6ht2qwhhb",
  "venue": "arXiv",
  "versions": []
}