{
  "abstract": "An induced matching is a matching that forms an induced subgraph. A graph is $t$-color-critical if removing some induced matching of size $t$ lowers its chromatic number, but removing any $t-1$ vertices does not. Let $F$ be a $t$-color-critical graph with $χ(F)=r+1$. For sufficiently large $n$, Simonovits determined the unique edge-extremal $F$-free graph on $n$ vertices. Recently, Zheng, Li and Li [Linear Algebra Appl.\\ 730 (2026) 546--565] conjectured that, for $t\\ge 2$ and $r\\ge 3$, the join $K_{t-1}\\vee T_{n-t+1,r}$ uniquely maximizes the signless Laplacian spectral radius among all $n$-vertex $F$-free graphs when $n$ is sufficiently large. In this paper, we prove this conjecture. In contrast to the usual spectral arguments, our proof of this conjecture relies on two techniques of a rather different flavour. Our first technique is an analogue of Zykov symmetrization for the signless Laplacian matrix. Our second technique is an induction on $n$, from which we obtain the lower bound on the smallest entry of the Perron vector of a signless Laplacian spectral extremal graph rather than a structural statement.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21367",
  "author_total": 4,
  "authors": [
    {
      "name": "Ming-Zhu Chen"
    },
    {
      "name": "Ya-Lei Jin"
    },
    {
      "name": "Peng-Li Zhang"
    },
    {
      "name": "Jian Zheng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7608yj6jtjehj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21367",
  "fulltext_url": null,
  "id": "7608yj6jtjehj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21367",
  "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": "W7213929356",
  "pdf_url": "https://arxiv.org/pdf/2609.21367",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7608yj6jtjehj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2609.21367",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21367"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A solution to a conjecture on the signless Laplacian spectral radius for $t$-color-critical graphs",
  "title_text": "A solution to a conjecture on the signless Laplacian spectral radius for t-color-critical graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/7608yj6jtjehj",
  "venue": "arXiv",
  "versions": []
}