  Fullerenes are molecules in the form of cage-like polyhedra, consisting
solely of carbon atoms. Fullerene graphs are mathematical models of fullerene
molecules. The transmission of a vertex $v$ of a graph is 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 $n \le 216$ are presented. 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.
