Fullerenes with the maximum Clar number

TL;DR

This paper investigates the Clar number in fullerenes, establishing bounds based on their order and characterizing those achieving maximum Clar numbers, with specific examples of experimentally produced fullerenes.

Contribution

It provides new bounds for the Clar number in fullerenes depending on their order modulo 6 and characterizes fullerenes attaining these bounds.

Findings

Fullerenes with order n ≡ 2 mod 6 do not reach the upper bound of Clar number.

Two specific fullerenes C80:1 and C80:2 attain the maximum Clar number bound.

Graph-theoretical characterization of fullerenes achieving maximum Clar numbers based on their order.

Abstract

The Clar number of a fullerene is the maximum number of independent resonant hexagons in the fullerene. It is known that the Clar number of a fullerene with n vertices is bounded above by [n/6]-2. We find that there are no fullerenes whose order n is congruent to 2 modulo 6 attaining this bound. In other words, the Clar number for a fullerene whose order n is congruent to 2 modulo 6 is bounded above by [n/6]-3. Moreover, we show that two experimentally produced fullerenes C80:1 (D5d) and C80:2 (D2) attain this bound. Finally, we present a graph-theoretical characterization for fullerenes, whose order n is congruent to 2 (respectively, 4) modulo 6, achieving the maximum Clar number [n/6]-3 (respectively, [n/6]-2).