{
  "abstract": "For fixed $r\\geq3$ and $0<α<1$, we study the maximum $α$-$s$-clique spectral radius among $n$-vertex $K_{r+1}$-free graphs. Here $\\mathcal A_{α,s}(G)=α\\mathcal D_s(G)+(1-α)\\mathcal A_s(G)$, where $\\mathcal A_s(G)$ and $\\mathcal D_s(G)$ are the adjacency and degree tensors of the $s$-clique hypergraph of $G$. For $s=r$, put $c_r=((r-1)/r)^{r-1}$. We prove that, for all sufficiently large $n$, the unique extremal graph is the Turán graph $T_r(n)$ if $α\\leq c_r$, and $K_1\\vee T_{r-1}(n-1)$ if $α>c_r$. For general $3\\leq s\\leq r$ and $n\\geq r$, every extremal graph is complete $r$-partite. The asymptotic maximum $α$-$r$-clique spectral radius among $n$-vertex $K_{r+1}$-free graphs is $\\max\\{r^{1-r},α(r-1)^{1-r}\\}n^{r-1}+o(n^{r-1})$. In every asymptotically extremal sequence of complete $r$-partite graphs, all parts have size $n/r+o(n)$ when $α<c_r$; when $α>c_r$, one part has size $o(n)$ and the remaining parts have size $n/(r-1)+o(n)$. At $α=c_r$, both part-size patterns are asymptotically extremal.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34196",
  "author_total": 2,
  "authors": [
    {
      "name": "Xiaoqi Liu"
    },
    {
      "name": "Haiying Shan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2mvhknmvgz3hm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34196",
  "fulltext_url": null,
  "id": "2mvhknmvgz3hm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34196",
  "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": "W7223848419",
  "pdf_url": "https://arxiv.org/pdf/2610.34196",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2mvhknmvgz3hm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34196",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34196"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The \\(\\mathcal A_α\\)-Spectral Turán Problem for \\(s\\)-Clique Tensors of \\(K_{r+1}\\)-Free Graphs",
  "title_text": "The 𝒜_α-Spectral Turán Problem for s-Clique Tensors of K_r+1-Free Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/2mvhknmvgz3hm",
  "venue": "arXiv",
  "versions": []
}