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On the classification of regular graphs with positive Lin-Lu-Yau curvature

  • George Brooks
  • Nathanael Johnson
  • William Linz
  • Xiaonan Liu
  • Linyuan Lu
  • Wynton Vaughn
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Abstract

We prove several new structural and classification results about d-regular graphs with positive Lin--Lu--Yau (LLY) curvature. We show that any positively curved d-regular graph has diameter at most 2d−2, which improves the previously best known diameter bound obtained from the Bonnet-Myers-type theorem for positively curved graphs. We further show that every positively curved d-regular graph with d≥ 3 is $3$-connected. We classify all positively curved $3$-regular graphs, as well as all positively curved regular planar graphs.

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Cite

BibTeXRIS
BibTeX
@misc{brooks2026classification,
  title = {{On the classification of regular graphs with positive Lin-Lu-Yau curvature}},
  author = {George Brooks and Nathanael Johnson and William Linz and Xiaonan Liu and Linyuan Lu and Wynton Vaughn},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08903},
  eprint = {2609.08903},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08903},
}
RIS
TY  - UNPB
TI  - On the classification of regular graphs with positive Lin-Lu-Yau curvature
AU  - George Brooks
AU  - Nathanael Johnson
AU  - William Linz
AU  - Xiaonan Liu
AU  - Linyuan Lu
AU  - Wynton Vaughn
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.08903
UR  - https://arxiv.org/abs/2609.08903
AB  - We prove several new structural and classification results about d-regular graphs with positive Lin--Lu--Yau (LLY) curvature. We show that any positively curved d-regular graph has diameter at most 2d−2, which improves the previously best known diameter bound obtained from the Bonnet-Myers-type theorem for positively curved graphs. We further show that every positively curved d-regular graph with d≥ 3 is $3$-connected. We classify all positively curved $3$-regular graphs, as well as all positively curved regular planar graphs.
ER  -