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)= Sigma u is an element of V(G)Sigma(s)(i=r)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 <= 100 (n <= 136) and small r and s. The irregular fullerene graphs are also presented.