Binomial-combinatorial properties of Clar structures

Abstract A Clar structure is defined to be a maximal indepenent set of vertices of the Clar graph of the corresponding benzenoid hydrocarbon. A special type of coloring, called a Clar coloring which is a specific type of 2-coloring is applied to Clar structures of two types of benzenoid hydrocarbons, viz., (a) nonbranched all-benzenoid systems, and (b) necklace-type hydrocarbons. It is found that the Clar counts of these two systems define what may be termed a ‘delayed’ Fibonacci sequence, in contrast to their Kekule counts which form a fibonacci sequence. Several combinatorial properties of Clar counts are given (eqs. (3)–(30)). The results emphasize the advantage of using Clar structures as bases of the (recently proposed [13]) valence-bond Hamiltonian instead of Kekule structures.