The maximum Wiener index of maximal planar graphs

The Wiener index of a connected graph is the sum of the distances between all pairs of vertices in the graph. It was conjectured that the Wiener index of an n -vertex maximal planar graph is at most $$\lfloor \frac{1}{18}(n^3+3n^2)\rfloor $$ ⌊ 1 18 ( n 3 + 3 n 2 ) ⌋ . We prove this conjecture and determine the unique n -vertex maximal planar graph attaining this maximum, for every $$ n\ge 10$$ n ≥ 10 .

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