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

Abstract

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) = u∈V (G) 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.