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On the Wiener Complexity and the Wiener Index of Fullerene Graphs

by Andrey A. Dobrynin 1,2,* and Andrei Yu Vesnin 1,2,3,*
Laboratory of Topology and Dynamics, Novosibirsk State University, 630090 Novosibirsk, Russia
Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, 630090 Novosibirsk, Russia
Regional Scientific and Educational Mathematical Center, Tomsk State University, 634050 Tomsk, Russia
Authors to whom correspondence should be addressed.
Mathematics 2019, 7(11), 1071;
Received: 16 October 2019 / Revised: 1 November 2019 / Accepted: 5 November 2019 / Published: 7 November 2019
(This article belongs to the Special Issue Graph Theory at Work in Carbon Chemistry)
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.
Keywords: graph; fullerene; Wiener index; Wiener complexity graph; fullerene; Wiener index; Wiener complexity
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Dobrynin, A.A.; Vesnin, A.Y. On the Wiener Complexity and the Wiener Index of Fullerene Graphs. Mathematics 2019, 7, 1071.

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