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

TL;DR

This paper investigates the Wiener (r,s)-complexity of fullerene graphs, analyzing their vertex transmission values, counting maximal complexity cases for various sizes, and exploring irregular fullerene structures.

Contribution

It introduces the concept of Wiener (r,s)-complexity for fullerene graphs and provides counts of maximal complexity cases up to certain sizes, including irregular structures.

Findings

Maximal Wiener (r,s)-complexity graphs are identified for n ≤ 100 (136).

Irregular fullerene graphs with high complexity are presented.

The Wiener (r,s)-complexity varies with graph size and parameters r, s.

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 $v$: $tr_{r,s}(v)= \sum_{u \in V(G)} \sum_{i=r}^{s} d(v,u)^i$ 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.