An upper bound on Wiener Indices of maximal planar graphs

• Published in: Discrete Applied Mathematics Vol. 258; pp. 76 - 86
• Main Authors: Che, Zhongyuan, Collins, Karen L.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 15.04.2019; Elsevier BV
• Subjects: Apexes; Apollonian network; Average distance; Graph theory; Graphs; Maximal planar graph; Total distance; Transmission; Upper bounds; Wiener index; Average distance; Maximal planar graph; Wiener index; Apollonian network; Total distance; Transmission
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2018.11.026

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.