{
  "abstract": "Classical Turán theory shows that, without additional constraints, the extremal structure of $F$-free graphs is governed by the chromatic number of $F$ and is asymptotically the balanced Turán graph. We investigate how prescribing a large maximum degree changes this picture and leads to an unbalanced Turán problem. Let $F$ be a graph with $χ(F)=r+1\\ge3$, and let $\\lceil(r-1)n/r\\rceil\\leΔ\\le n-1$. We study the maximum number of edges and adjacency spectral radius of an $n$-vertex $F$-free graph with maximum degree exactly $Δ$. For $F=K_{r+1}$, the unbalanced $r$-partite graph $S_{n,Δ}^{(r)}=(n-Δ)K_1\\vee T(Δ,r-1)$ is the unique maximizer of both quantities. Let $a(F)$ be the minimum size of an independent set $I$ such that $χ(F-I)\\le r$. If $a(F)=1$, we prove edge and spectral stability with respect to $S_{n,Δ}^{(r)}$. If $a(F)>1$, the extremal values are $t(n,r)+o(n^2)$ and $ρ(T(n,r))+o(n)$, respectively, and every extremal graph differs from $T(n,r)$ in $o(n^2)$ edges, uniformly all $\\lceil(r-1)n/r\\rceil\\leΔ\\le n-1$. We further establish a general spectral transfer principle: for finite forbidden families, a decomposition-family bound of order $O(n^{1+s})$ yields a spectral bound with error $O(n^s)$ for $0\\le s<1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.26634",
  "author_total": 1,
  "authors": [
    {
      "name": "Chang Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/br9fakhqsbmvy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.26634",
  "fulltext_url": null,
  "id": "br9fakhqsbmvy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.26634",
  "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": "W7204524840",
  "pdf_url": "https://arxiv.org/pdf/2608.26634",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/br9fakhqsbmvy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2608.26634",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.26634"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Unbalanced Turán and spectral Turán problems with prescribed large maximum degree",
  "title_text": "Unbalanced Turán and spectral Turán problems with prescribed large maximum degree",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/br9fakhqsbmvy",
  "venue": "arXiv",
  "versions": []
}