{
  "abstract": "We establish an entropy bridge, and then use it to prove that for any color-critical graph $F$ with chromatic number $χ(F)=r+1\\ge 3$, there exists a constant $λ_0=λ_0(F)$ such that if $G$ is an $F$-free graph with $λ(G)\\ge λ_0$, then for every $\\ell\\ge 1$ with $(r,\\ell )\\neq (2,2)$, \\[ λ^\\ell (G) \\le \\Big(1-\\frac1r\\Big)w_\\ell(G), \\] with equality if and only if $G$ is a regular complete $r$-partite graph; in the case $r=2$ with $\\ell$ even, equality holds for every complete bipartite graph. The pair $(r,\\ell )=(2,2)$ must be excluded, since the bound fails for several forbidden graphs, e.g., $C_{2t+1}$ with $t\\ge 2$. Moreover, walks counts may also be replaced by the homomorphism counts of unbalanced trees. As further applications, we extend a spectral supersaturation result of Bollobás and Nikiforov [J. Combin. Theory Ser. B (2007)], and we also extend the entropic Turán theorem of Chao and Yu [J. London Math. Soc. (2026)] from $K_{r+1}$-free graphs to $F$-free graphs with $F$ color-critical. We provide a framework by passing through weighted Turá theorems of independent interest. If $G$ is $F$-free and $\\mathbf{p}$ is a probability vector on $V(G)$ with $\\lVert \\mathbf{p}\\rVert_\\infty$ sufficiently small, then \\[ 2\\sum_{uv\\in E(G)}p_up_v \\le 1-\\frac1r + o(1), \\] and the error term $o(1)$ can be removed if and only if $F$ is color-critical. This is a Motzkin-Straus-type inequality in which the clique number of $G$ is replaced by $χ(F)-1$. The proof of this weighted result combines a blow-up argument, the graph removal lemma, the Erdős-Simonovits stability theorem, a probabilistic sampling argument, and an exact estimate near a complete $r$-partite graph. The bridge linking spectral inequalities to weighted inequalities is based on the entropy method for the Markov chain attached to the Perron vector.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.24388",
  "author_total": 1,
  "authors": [
    {
      "name": "Yongtao Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e0jqde6k58p3v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.24388",
  "fulltext_url": null,
  "id": "e0jqde6k58p3v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.24388",
  "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": "W7204232751",
  "pdf_url": "https://arxiv.org/pdf/2608.24388",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e0jqde6k58p3v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2608.24388",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.24388"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "An entropy bridge from weighted to spectral Turán theorems",
  "title_text": "An entropy bridge from weighted to spectral Turán theorems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/e0jqde6k58p3v",
  "venue": "arXiv",
  "versions": []
}