{
  "abstract": "The binding number $b(G)$ of a graph $G$ is the minimum value of $|N_{G}(X)|/|X|$ taken over all nonempty sets $X\\subseteq V(G)$ such that $N_{G}(X)\\neq V(G)$. A graph $G$ is 1-binding if $b(G)\\geq 1$. Fan and Lin [Electron. J. Combin. 31 (2024) P1.30] characterized the unique spectral extremal graphs among 1-binding graphs with $δ(G)\\geq k$ that contain no $k$-factor for $k=1$, $2$. Subsequently, Tang and Zhang solved the problem for $k\\geq 3$. Note that $δ(G)\\geq a$ is a necessary condition for the existence of an $[a,b]$-factor in a graph $G$. For $1\\leq a<b$, Fan et al. [Discrete Math. 349 (2026) 115103] determined the unique spectral extremal graph $H_n^{a,b}$ among connected graphs with $δ(G)\\geq a$ that contain no $[a,b]$-factor. Note that $b(H_n^{a,b})<1$. This motivates the corresponding spectral extremal problem for 1-binding graphs. Note that every 1-binding graph with $δ(G)\\geq 1$ contains a $[1,b]$-factor for $b>1$. For $2\\leq a<b$, we in this paper determine the unique extremal graph $\\mathcal{B}_n^{a,b}$ with the maximum spectral radius among 1-binding graphs of order $n$ with $δ(G)\\geq a$ that contain no $[a,b]$-factor. The toughness of a non-complete graph $G$ is $t(G)=\\min\\{\\frac{|X|}{c(G-X)}\\mid X\\subseteq V(G)\\, \\text{and}\\, c(G-X)\\geq 2\\}$. A graph $G$ is 1-tough if $c(G-X)\\leq |X|$ for every $X\\subseteq V(G)$ with $c(G-X)\\geq 2$. Every 1-tough graph of order $n\\geq2$ is 1-binding. Note that $t(\\mathcal{B}_n^{a,b})<1$. This leads us to investigate the corresponding spectral extremal problem for 1-tough graphs. Note that every 1-tough graph with $δ(G)\\geq 1$ contains a $[1,b]$-factor for $b>1$. For $2\\leq a<b$, we in this paper characterize the unique extremal graph with the maximum spectral radius among 1-tough graphs of order $n$ with $δ(G)\\geq a$ containing no $[a,b]$-factor.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16712",
  "author_total": 2,
  "authors": [
    {
      "name": "Ting Xu"
    },
    {
      "name": "Ruifang Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/crz0sssnte2n8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16712",
  "fulltext_url": null,
  "id": "crz0sssnte2n8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16712",
  "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": "W7221754309",
  "pdf_url": "https://arxiv.org/pdf/2610.16712",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/crz0sssnte2n8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.16712",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16712"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Spectral extremal problems on $[a,b]$-factors in binding graphs, and beyond",
  "title_text": "Spectral extremal problems on [a,b]-factors in binding graphs, and beyond",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/crz0sssnte2n8",
  "venue": "arXiv",
  "versions": []
}