In this paper the Clar covering polynomial of a hexagonal system is introduced. In fact it is a kind of F polynomial [4] of a graph, and can be calculated by recurrence relations. We show that the number of aromatic sextets (in a Clar formula), the number of Clar formulas, the number of Kekule structures and the first Herndon number for any Kekulean hexagonal system can be easily obtained by it...

The Clar number of a (hydro)carbon molecule, introduced by Clar [E. Clar, The aromatic sextet, (1972).], is the maximum number of mutually disjoint resonant hexagons in the molecule. Calculating the Clar number can be formulated as an optimization problem on 2-connected planar graphs. Namely, it is the maximum number of mutually disjoint even faces a perfect matching can simultaneously alternat...

A Clar set of a benzenoid graph B is a maximum set of independent alternating hexagons over all perfect matchings of B. The Clar number of B, denoted by Cl(B), is the number of hexagons in a Clar set for B. In this paper, we first prove some results on the independence number of subcubic trees to study the Clar number of catacondensed benzenoid graphs. As the main result of the paper we prove a...

An explicit combinatorial expression is obtained for the Zhang-Zhang polynomial (also known as “Clar cover polynomial”) of a large class of pericondensed benzenoid systems, the multiple linear hexagonal chains Mn,m. By means of this result, various problems encountered in the Clar theory of Mn,m are also resolved: counting of Clar and Kekulé structures, determining the Clar number, and calculat...

A fullerene graph is a cubic 3-connected plane graph with (exactly 12) pentagonal faces and hexagonal faces. Let Fn be a fullerene graph with n vertices. A set H of mutually disjoint hexagons of Fn is a sextet pattern if Fn has a perfect matching which alternates on and off each hexagon in H. The maximum cardinality of sextet patterns of Fn is the Clar number of Fn. It was shown that the Clar n...

Clar's aromatic sextet theory provides a good means to describe the aromaticity of benzenoid hydrocarbons, which was mainly based on experimental observations. Clar defined sextet pattern and Clar number of benzenoid hydrocarbons, and he observed that for isomeric benzenoid hydrocarbons, when Clar number increases the absorption bands shift to shorter wavelength, and the stability of these isom...

It is shown that the number of Clar formulas of a Kekuléan benzenoid system B is equal to the number of subgraphs of the resonance graph of B isomorphic to the Cl(B)-dimensional hypercube, where Cl(B) is the Clar number of B.