On the Wiener (r,s)-complexity of fullerene graphs

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.