Discoveries / combinatorics

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 $d=4$.