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

The Maximum Wiener Index of Maximal Planar Graphs