The Clar Covering Polynomial of Hexagonal Systems I

Abstract 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 its Clar covering polynomial. In addition, we give some theorems to calculate the Clar covering polynomial of a hexagonal system. As examples we finally derive the explicit expressions of the Clar covering polynomials for some small hexagonal systems and several types of catacondensed hexagonal systems. A relation between the resonance energy and the Clar covering polynomial of a hexagonal system is considered in the next paper.

[1]  Horst Sachs,et al.  Remark on the Dimer Problem , 1994, Discret. Appl. Math..

[2]  Heping Zhang,et al.  The Clar covering polynomial of hexagonal systems with an application to chromatic polynomials , 1997, Discret. Math..

[3]  Zhi-bo Chen,et al.  Directed tree structure of the set of kekulé patterns of polyhex graphs , 1985 .

[4]  Pierre Hansen,et al.  Upper bounds for the Clar number of a benzenoid hydrocarbon , 1992 .

[5]  Zheng Mao-lin,et al.  A maximal cover of hexagonal systems , 1985 .

[6]  Ivan Gutman,et al.  A regularity in Herndon's resonance energy method , 1985 .

[7]  Zhang He-ping,et al.  The clar formulas of regular t-tier strip benzenoid systems , 1995 .

[8]  W. C. Herndon.  THERMOCHEMICAL PARAMETERS FOR BENZENOID HYDROCARBONS , 1974 .

[9]  Guo Xiaofeng,et al.  Directed tree structure of the set of Kekule´ patterns of generalized polyhex graphs , 1991 .

[10]  S. J. Cyvin,et al.  Introduction to the theory of benzenoid hydrocarbons , 1989 .

[11]  Ioan Tomescu,et al.  Alternating 6-cycles in perfect matchings of graphs representing condensed benzenoid hydrocarbons , 1988, Discret. Appl. Math..

[12]  E. Clar.  The aromatic sextet , 1972 .

[13]  S. J. Cyvin,et al.  Kekule Structures in Benzenoid Hydrocarbons , 1988 .

[14]  Fuji Zhang,et al.  When each hexagon of a hexagonal system covers it , 1991, Discret. Appl. Math..

[15]  Haruo Hosoya,et al.  On some counting polynomials in chemistry , 1988, Discret. Appl. Math..

[16]  Haruo Hosoya,et al.  Sextet polynomial. A new enumeration and proof technique for the resonance theory applied to the aromatic hydrocarbons. , 1975 .

[17]  P. Hansen,et al.  Normal components of benzenoid systems , 1993 .