On the Wiener Complexity and the Wiener Index of Fullerene Graphs

Andrey A Dobrynin,Andrei Yu Vesnin

doi:10.3390/math7111071

Abstract

Fullerenes are molecules that can be presented in the form of cage-like polyhedra, consisting only of carbon atoms. Fullerene graphs are mathematical models of fullerene molecules. The transmission of a vertex v of a graph is a local graph invariant defined as the sum of distances from v to all the other vertices. The number of different vertex transmissions is called the Wiener complexity of a graph. Some calculation results on the Wiener complexity and the Wiener index of fullerene graphs of order n ≤ 232 and IPR fullerene graphs of order n ≤ 270 are presented. The structure of graphs with the maximal Wiener complexity or the maximal Wiener index is discussed, and formulas for the Wiener index of several families of graphs are obtained.

Highlights

A fullerene is a spherically shaped molecule consisting of carbon atoms in which every carbon ring forms a pentagon or a hexagon
We present some results of studies of the Wiener complexity and the Wiener index of fullerene graphs
Does there exist an IPR fullerene graph with maximal Wiener complexity Cn? We believe that the answer to this question will be positive for sufficiently large n

Summary

Introduction

A fullerene is a spherically shaped molecule consisting of carbon atoms in which every carbon ring forms a pentagon or a hexagon. The set of fullerene graphs with n vertices will be denoted as Fn. The number of faces of graphs in Fn is f = n/2 + 2 and, the number of hexagonal faces is n/2 − 10. The number of different vertex transmissions in a graph G is known as the Wiener complexity [43] (or the Wiener dimension [7]), CW ( G ). A graph is called transmission irregular if all vertices of the graph have pairwise different transmissions, i.e., it has the largest possible Wiener complexity. We are interested in two questions: does a transmission irregular fullerene graph exist and can a graph with the maximal Wiener complexity has the maximal Wiener index?

Wiener Complexity of Fullerene Graphs

Graphs with the Maximal Wiener Complexity

Fullerene Graphs with the Maximal Wiener Index

IPR Fullerene Graphs with the Maximal Wiener Index

Conclusions