{
  "abstract": "Let $\\textit{spex}_{\\mathcal{P}_1}(n,F)$ be the maximum spectral radius among all $n$-vertex $F$-free $1$-planar graphs. Define $F_t$ as the friendship graph formed by $t$ triangles sharing exactly one common vertex. Tait and Tobin (2017)~\\cite{Tait2017} used the fundamental structure of spectral extremal graphs to determine the unique planar graph with maximum spectral radius for sufficiently large order. Subsequently, Zhang, Wang and Wang (2024)~\\cite{Zhang2024} characterized the corresponding extremal graph in the class of $1$-planar graphs. In this paper, we focus on $F_t$-free $1$-planar graphs and establish a structural theorem for their spectral extremal graphs for all $t\\geq1$ and sufficiently large $n$. More precisely, every extremal graph is connected and contains a copy of $K_{2,n-2}$, and for $t\\geq2$ the two distinguished vertices are adjacent and the subgraph induced by the remaining vertices is a bipartite graph. Based on this structure result together with the drawing properties of $K_{3,6}$, we determine $\\textit{spex}_{\\mathcal{P}_1}(n,F_t)$ and characterize its unique extremal graph.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07805",
  "author_total": 3,
  "authors": [
    {
      "name": "Jiamin Li"
    },
    {
      "name": "Dan Li"
    },
    {
      "name": "Yuanyuan Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bdjwmw0nebn4g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07805",
  "fulltext_url": null,
  "id": "bdjwmw0nebn4g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07805",
  "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": "W7221041860",
  "pdf_url": "https://arxiv.org/pdf/2610.07805",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bdjwmw0nebn4g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.07805",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07805"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Spectral extremal problems on 1-planar graphs without Friendship graph",
  "title_text": "Spectral extremal problems on 1-planar graphs without Friendship graph",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/bdjwmw0nebn4g",
  "venue": "arXiv",
  "versions": []
}