Spectral Extremal 1-Planar Graphs with Bounded Pentagon Packing

Abstract

Let 𝒫₁ denote the class of 1-planar graphs and let tC₅ be the disjoint union of t copies of C₅. For every fixed t≥3 and all sufficiently large n, we determine the unique n-vertex tC₅-free graph in 𝒫₁ with maximum adjacency spectral radius, answering Problem 1 of Li, Wang and Zhao. The proof first gives a structural description of every extremizer. After two dominating vertices are removed, the remainder consists of copies of the seven-vertex graph B=K₁∨2K₃ together with at most one bounded connected core. This follows from a two-family covering theorem for 1-planar K₂-joins and the sharp packing--defect inequality 12v(Q)−7e(Q)≥1−10ν₅(Q). The same inequality yields an exact edge-extremal result for remainders with bounded pentagon packing. A normalized resolvent then cancels the repeated B-components, and finite moment comparisons force all packing and all nonzero defect into one core and identify that core uniquely in each residue class modulo $7$. The case t=3 is the pentagon-free boundary case and is completed by one exact finite component lemma.

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BibTeX RIS

BibTeX

@misc{zhang2026spectral,
  title = {{Spectral Extremal 1-Planar Graphs with Bounded Pentagon Packing}},
  author = {Zhanhe Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.08888},
  eprint = {2610.08888},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.08888},
}

RIS

TY  - UNPB
TI  - Spectral Extremal 1-Planar Graphs with Bounded Pentagon Packing
AU  - Zhanhe Zhang
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.08888
UR  - https://arxiv.org/abs/2610.08888
AB  - Let 𝒫_1 denote the class of 1-planar graphs and let tC_5 be the disjoint union of t copies of C_5. For every fixed t≥3 and all sufficiently large n, we determine the unique n-vertex tC_5-free graph in 𝒫_1 with maximum adjacency spectral radius, answering Problem 1 of Li, Wang and Zhao. The proof first gives a structural description of every extremizer. After two dominating vertices are removed, the remainder consists of copies of the seven-vertex graph B=K_1∨2K_3 together with at most one bounded connected core. This follows from a two-family covering theorem for 1-planar K_2-joins and the sharp packing--defect inequality 12v(Q)−7e(Q)≥1−10ν_5(Q). The same inequality yields an exact edge-extremal result for remainders with bounded pentagon packing. A normalized resolvent then cancels the repeated B-components, and finite moment comparisons force all packing and all nonzero defect into one core and identify that core uniquely in each residue class modulo $7$. The case t=3 is the pentagon-free boundary case and is completed by one exact finite component lemma.
ER  -

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