An upper bound on Wiener Indices of maximal planar graphs

Abstract

The Wiener index of a connected graph is the summation of distances between all unordered pairs of vertices of the graph. The status of a vertex in a connected graph is the summation of distances between the vertex and all other vertices of the graph. A maximal planar graph is a graph that can be embedded in the plane such that the boundary of each face (including the exterior face) is a triangle. Let G be a maximal planar graph of order n≥3. In this paper, we show that the diameter of G is at most ⌊13(n+1)⌋, and the status of a vertex of G is at most ⌊16(n2+n)⌋. Both of them are sharp bounds and can be realized by an Apollonian network, which is a chordal maximal planar graph. We also present a sharp upper bound ⌊118(n3+3n2)⌋ on Wiener indices when graphs in consideration are Apollonian networks of order n≥3. We further show that this sharp upper bound holds for maximal planar graphs of order 3≤n≤10, and conjecture that it is valid for all n≥3.