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Improved Bounds on the Szeged-Wiener Gap and the BKLPS Conjecture

  • Lily Zhang
  • Evan Li
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Abstract

Bonamy-Knor-Lužar-Pinlou-Škrekovski (2017) define Knt to be the complete graph of n−1 vertices but with an extra vertex that's adjacent to t vertices of the complete graph part. They propose a stronger conjecture which asserts that if G is a finite simple $2$-connected graph of order n ≥ 10 not isomorphic to Kn, Kn2, nor Knn−2, then the Szeged-Wiener gap of G is η(G) ≥ 2n. We improve upon their work to tighten the bounds on the Szeged-Wiener gap, allowing us to prove this conjecture in the affirmative. Afterwards, we construct graphs attaining equality for each n ≥ 10 and pose a problem for interested readers to determine a necessary and sufficient condition for equality.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026improved,
  title = {{Improved Bounds on the Szeged-Wiener Gap and the BKLPS Conjecture}},
  author = {Lily Zhang and Evan Li},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20025},
  eprint = {2609.20025},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20025},
}
RIS
TY  - UNPB
TI  - Improved Bounds on the Szeged-Wiener Gap and the BKLPS Conjecture
AU  - Lily Zhang
AU  - Evan Li
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.20025
UR  - https://arxiv.org/abs/2609.20025
AB  - Bonamy-Knor-Lužar-Pinlou-Škrekovski (2017) define K_n^t to be the complete graph of n−1 vertices but with an extra vertex that's adjacent to t vertices of the complete graph part. They propose a stronger conjecture which asserts that if G is a finite simple $2$-connected graph of order n ≥ 10 not isomorphic to K_n, K_n^2, nor K_n^n−2, then the Szeged-Wiener gap of G is η(G) ≥ 2n. We improve upon their work to tighten the bounds on the Szeged-Wiener gap, allowing us to prove this conjecture in the affirmative. Afterwards, we construct graphs attaining equality for each n ≥ 10 and pose a problem for interested readers to determine a necessary and sufficient condition for equality.
ER  -