  Fullerene graphs are mathematical models of fullerene molecules. The Wiener
$(r,s)$-complexity of a fullerene graph $G$ with vertex set $V(G)$ is the
number of pairwise distinct values of $(r,s)$-transmission $tr_{r,s}(v)$ of its
vertices $v$: $tr_{r,s}(v)= \sum_{u \in V(G)} \sum_{i=r}^{s} d(v,u)^i$ for
positive integer $r$ and $s$. The Wiener $(1,1)$-complexity is known as the
Wiener complexity of a graph. Irregular graphs have maximum complexity equal to
the number of vertices. No irregular fullerene graphs are known for the Wiener
complexity. Fullerene (IPR fullerene) graphs with n vertices having the maximal
Wiener $(r,s)$-complexity are counted for all $n\le 100$ ($n\le 136$) and small
$r$ and $s$. The irregular fullerene graphs are also presented.
