Spectral extremal problems on 1-planar graphs without Friendship graph

Abstract

Let spex_𝒫₁(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≥1 and sufficiently large n. More precisely, every extremal graph is connected and contains a copy of K_2,n−2, and for t≥2 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 spex_𝒫₁(n,F_t) and characterize its unique extremal graph.

Spectral Extremal 1-Planar Graphs with Bounded Pentagon Packing
Zhanhe Zhang
arXiv6 Oct 2026PreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
Spectral extremal hypergraphs without long Berge cycles
Lihua Feng, Lu Lu, Tingzeng Wu
arXiv1 Sep 2026PreprintCC0Full textCombinatorics & discrete mathematicsarXiv
Spectral Extremal Graphs without a K_k-Factor
Cunxiang Duan, Tingting Han, Lin-Peng Zhang
arXiv18 Sep 2026PreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
Extremal spectral result of outerplanar graphs without P_{3· l}
Fulong Ye, Yuxiang Liu, Ligong Wang
arXiv21 Sep 2026PreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
Spectral extremal problems on [a,b]-factors in binding graphs, and beyond
Ting Xu, Ruifang Liu
arXiv12 Oct 2026PreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv

Cite

BibTeX RIS

BibTeX

@misc{li2026spectral,
  title = {{Spectral extremal problems on 1-planar graphs without Friendship graph}},
  author = {Jiamin Li and Dan Li and Yuanyuan Chen},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.07805},
  eprint = {2610.07805},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.07805},
}

RIS

TY  - UNPB
TI  - Spectral extremal problems on 1-planar graphs without Friendship graph
AU  - Jiamin Li
AU  - Dan Li
AU  - Yuanyuan Chen
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.07805
UR  - https://arxiv.org/abs/2610.07805
AB  - Let spex_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≥1 and sufficiently large n. More precisely, every extremal graph is connected and contains a copy of K_2,n−2, and for t≥2 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 spex_𝒫_1(n,F_t) and characterize its unique extremal graph.
ER  -

Abstract

Related

Cite