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 $t{r}_{r,s}(v)$ of its vertices v: $t{r}_{r,s}(v)=\sum _{u\in V(G)}{\sum }_{i=r}^{s}d{(v,u)}^i$ 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.

Keywords:

• Fullerenes
• transmission
• Wiener complexity

Additional information

Funding

The work was supported by the Russian Foundation for Basic Research (project number 19-01-00682), the state contract of the Sobolev Institute of Mathematics (project no. 0314-2019-0016) (AAD), and the Ministry of Science and Education of Russia (agreement no. 075-02-2021-1392) (AV).