The DeLaViña–Waller conjecture on the Wiener index

Every finite simple connected graph $G$ with$$|V(G)|=2d+1,\qquad \operatorname{diam}(G)=d\ge 3$$satisfies$$W(G)\le W(C_{2d+1}) =\frac{(2d+1)d(d+1)}2.$$The claimed equality cases are exactly $C_{2d+1}$ for every $d\ge3$, the double star $D_{2,3}$ when $d=3$, and the nine-vertex tree $T_{1,2,2}=S(2,3,3)$ when $d=4$.

Result

Proved

Status

Candidate (review pending)

AI contribution

AI co-developed

Method

Argument

Field

Extremal Graph Theory

Posed by

E. DeLaViña and B. Waller

Year posed

2008

Years open

18y

Solved

2026-08-19

Model

GPT-5.6 Sol; Claude Fable 5

Vendor

OpenAI, Anthropic

Collaborators

Mingchang Liu

Verification

Unreviewed

Publication

Preprint

Significance

10 / 100*

Disclosed cost

—

Wikipedia

No dedicated article

Submitted by SilentIbis759 on 20 Aug 2026