The Maximum Wiener Index of Maximal Planar Graphs

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 $n$-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 for which this maximum is attained.