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PreprintCC BY 4.0arXiv

A proof of the resistance diameter conjecture for line graphs

  • Xiang-Feng Pan
  • Xiang-Yang Liu
  • Zhen-Mu Hong
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Abstract

We prove the conjecture of Xu, Li, Hua, and Pan that the resistance diameter does not increase under the line-graph operation. For every finite connected simple graph G with at least one edge, we establish Dr(L(G))≤ Dr(G), with equality if and only if G is a cycle or K4. The proof combines an exact electrical representation of L(G) by stars with a sharp budget inequality for the branch core. This inequality controls the combined error terms arising from the comparison of degree-two paths and is central to both the diameter bound and the equality analysis.

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Cite

BibTeXRIS
BibTeX
@misc{pan2026proof,
  title = {{A proof of the resistance diameter conjecture for line graphs}},
  author = {Xiang-Feng Pan and Xiang-Yang Liu and Zhen-Mu Hong},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.15133},
  eprint = {2609.15133},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.15133},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A proof of the resistance diameter conjecture for line graphs
AU  - Xiang-Feng Pan
AU  - Xiang-Yang Liu
AU  - Zhen-Mu Hong
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.15133
UR  - https://arxiv.org/abs/2609.15133
AB  - We prove the conjecture of Xu, Li, Hua, and Pan that the resistance diameter does not increase under the line-graph operation. For every finite connected simple graph G with at least one edge, we establish D_r(L(G))≤ D_r(G), with equality if and only if G is a cycle or K_4. The proof combines an exact electrical representation of L(G) by stars with a sharp budget inequality for the branch core. This inequality controls the combined error terms arising from the comparison of degree-two paths and is central to both the diameter bound and the equality analysis.
ER  -