The Maximum Wiener Index of Maximal Planar Graphs
Debarun Ghosh, Ervin Gy\H{o}ri, Addisu Paulos, Nika Salia, Oscar, Zamora

TL;DR
This paper proves a conjecture that the Wiener index of any n-vertex maximal planar graph is bounded above by a specific cubic function and identifies the unique extremal graph for each n ≥ 10.
Contribution
It confirms the conjectured maximum Wiener index for maximal planar graphs and characterizes the extremal graphs achieving this maximum.
Findings
The Wiener index of n-vertex maximal planar graphs is at most (n^3+3n^2).
The extremal graphs attaining this maximum are uniquely determined for each n .
The conjecture is rigorously proven for all n , n , with explicit extremal examples.
Abstract
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 -vertex maximal planar graph is at most . We prove this conjecture and for every , , determine the unique -vertex maximal planar graph for which this maximum is attained.
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