Graph-theoretical analysis of the clar's aromatic sextet : Mathematical properties of the set of the kekulé patterns and the sextet polynomial for polycyclic aromatic hydrocarbons

Abstract

The mathematical structure of a set of the Kekulé patterns for a polycyclic aromatic hydrocarbon has been analysed graph-theoretically. By defining the proper and improper sextets, sextet pattern, Clar transformation, and sextet rotation, one can prove the important property of the sextet polynomial B G(x) as B G(1) = K(G), where K(G) is the number of the Kekulé patterns for thin polyhex graph G. For fat polyhex graphs such as coronene the above relation is found to be also valid by introducing the concept of a super sextet. All the Kekule patterns for a given G are shown to form a hierarchical tree structure by the sextet rotation. The theory developed in this paper gives a mathematical basis and interpretation for the concept of the Clar's aromatic sextet.