An upper bound on Wiener Indices of maximal planar graphs

Zhongyuan Che, Karen L. Collins

Research output: Contribution to journalArticle

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 ⌊[Formula presented](n+1)⌋ and the status of a vertex of G is at most ⌊[Formula presented](n 2 +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 ⌊[Formula presented](n 3 +3n 2 )⌋ 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.

Original languageEnglish (US)
Pages (from-to)76-86
Number of pages11
JournalDiscrete Applied Mathematics
Volume258
DOIs
StatePublished - Apr 15 2019

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Wiener Index
Planar graph
Upper bound
Graph in graph theory
Summation
Connected graph
Vertex of a graph
Face
Unordered
Sharp Bound
Triangle
Valid

All Science Journal Classification (ASJC) codes

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

Cite this

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An upper bound on Wiener Indices of maximal planar graphs. / Che, Zhongyuan; Collins, Karen L.

In: Discrete Applied Mathematics, Vol. 258, 15.04.2019, p. 76-86.

Research output: Contribution to journalArticle

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