The DeLaViña-Waller conjecture on the Wiener index
Every connected graph on 2d+1 vertices of diameter d at least 3 is claimed to have Wiener index at most that of the cycle on the same number of vertices, with the equality cases determined.
- Lab
- Independent
- Model
- GPT-5.6 Sol; Claude Fable 5
- Field
- Mathematics
- Date
- 2026-08-19
- Human collaborators
- Mingchang Liu
- Problem posed
- 2008 · open 18 yrs
Sources
Original work
Independent commentary
What was found
The conjecture of DeLaViña and Waller states that a finite simple connected graph on 2d+1 vertices with diameter exactly d at least 3 has Wiener index, the sum of all pairwise distances, at most that of the odd cycle on 2d+1 vertices. The preprint claims a proof and pins down the equality cases: the odd cycle for every d at least 3, plus the double star D(2,3) when d = 3 and the nine-vertex tree S(2,3,3) when d = 4.
Novelty check
The conjecture is named, attributed to DeLaViña and Waller and dated 2008 in the vibemathed record, which is where this candidate is carried. The claim is sharpened rather than merely asserted, since the equality cases are enumerated and include two sporadic small-diameter graphs alongside the cycle, and those exceptions are the sort of detail a re-derivation can check. No prior proof appears in that record.
Caveats and known objections
A Zenodo preprint, unrefereed, not formalized, with no independent check on record, and vibemathed lists it as a candidate pending review with a low significance rating. The Zenodo metadata carries no abstract, so the statement here is reconstructed from the registry record rather than from the deposit's own description, and a reader should open the deposit before relying on it. Autonomy is graded ai-assisted on the only account of the model's role available, which is the registry's one-line summary that the models assisted in developing proof strategies and checking computations; no AI disclosure written by the author has been located.
Also recorded at
vibemathedthe-delavina-waller-conjecture-on-the-wiener-index
Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.
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Entry history (1 event)
- AddedEntered the registry graded Author verified and AI-assisted.
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Cite this entry
whataifound.org. (2026). The DeLaViña-Waller conjecture on the Wiener index. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-08-19-delavina-waller-wienerBibTeX
@misc{whataifound-independent-2026-wiener,
title = {The DeLaViña-Waller conjecture on the Wiener index},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Independent. Verification: Author verified. Autonomy: AI-assisted.},
url = {https://whataifound.org/finding/2026-08-19-delavina-waller-wiener}
}