### 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 language | English (US) |
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Pages (from-to) | 76-86 |

Number of pages | 11 |

Journal | Discrete Applied Mathematics |

Volume | 258 |

DOIs | |

State | Published - Apr 15 2019 |

### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Applied Mathematics

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## Cite this

*Discrete Applied Mathematics*,

*258*, 76-86. https://doi.org/10.1016/j.dam.2018.11.026