### 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) |
---|---|

Pages (from-to) | 76-86 |

Number of pages | 11 |

Journal | Discrete Applied Mathematics |

Volume | 258 |

DOIs | |

State | Published - Apr 15 2019 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Discrete Applied Mathematics*,

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

}

*Discrete Applied Mathematics*, vol. 258, pp. 76-86. https://doi.org/10.1016/j.dam.2018.11.026

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

Research output: Contribution to journal › Article

TY - JOUR

T1 - An upper bound on Wiener Indices of maximal planar graphs

AU - Che, Zhongyuan

AU - Collins, Karen L.

PY - 2019/4/15

Y1 - 2019/4/15

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85058562374&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85058562374&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2018.11.026

DO - 10.1016/j.dam.2018.11.026

M3 - Article

VL - 258

SP - 76

EP - 86

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

ER -