The maximum Wiener index of maximal planar graphs

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 $$\lfloor \frac{1}{18}(n^3+3n^2)\rfloor $$ ⌊ 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 .

Proximity in triangulations and quadrangulations

“…This paper is concerned with bounds on the proximity of triangulations, i.e., maximal planar graphs, and quadrangulations, i.e., maximal bipartite planar graphs. Several bounds on distance measures in maximal planar graphs are known, for example for radius [1], average eccentricity [2], Wiener index [10,11,16,17]. (The Wiener index of a graph is the sum of the distances between all unordered pairs of graphs.…”