# On the Wiener (r,s)-complexity of fullerene graphs: Fullerenes, Nanotubes and Carbon Nanostructures: Vol 0, No 0

## Description

Summary

Fullerene graphs are mathematical fashions of fullerene molecules. The Wiener (r,s)-complexity of a fullerene graph G with vertex set V(G) is the variety of pairwise distinct values of (r,s)-transmission

$trr,s(v)$

of its vertices v:

$trr,s(v)=∑u∈V(G)∑i=rsd(v,u)i$

for constructive integer r and s. The Wiener (1,1)-complexity is named the Wiener complexity of a graph. Irregular graphs have most complexity equal to the variety of vertices. No irregular fullerene graphs are identified for the Wiener complexity. Fullerene (IPR fullerene) graphs with n vertices having the maximal Wiener (r,s)-complexity are counted for all

$n≤100$

(

$n≤136$

) and small r and s. The irregular fullerene graphs are additionally introduced.