Extremal spectral result of outerplanar graphs without P_{3· l}

@misc{ye2026extremal,
  title = {{Extremal spectral result of outerplanar graphs without $P_3\cdot l$}},
  author = {Fulong Ye and Yuxiang Liu and Ligong Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.24080},
  eprint = {2609.24080},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24080},
}

TY  - UNPB
TI  - Extremal spectral result of outerplanar graphs without P_3· l
AU  - Fulong Ye
AU  - Yuxiang Liu
AU  - Ligong Wang
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.24080
UR  - https://arxiv.org/abs/2609.24080
AB  - A graph G is F-free if it does not contain F as a subgraph. Let spex(n,F) be the maximum spectral radius over all n-vertex F-free outerplanar graphs. For integers t≥1 and l≥2, let P_t· l be the starlike tree with t branches of length l−1. For sufficiently large n, Yin, Li, and Meng [arXiv:2504.04364v1] characterized the unique extremal graph for spex(n,P_t· l) when t=1, t=2, or t≥4. They left the case t=3 open and proposed a natural candidate for the extremal graph. We show that this candidate is not extremal and determine the unique extremal graph for spex(n,P_3· l). For every l≥3 and all sufficiently large n, this unique extremal graph is K_1∨(2P_2l−3∪ qP_l−2∪ P_r), where q and r are integers satisfying n=2(2l−3)+q(l−2)+r+1, q≥0, 0≤ r<l−2.
ER  -