Clar polynomials

Randic, M., El-Basil, S., Nikolic, S. and Trinajstic, N. (1998c). Clar Polynomials of Large Benze-noid Systems. J.Chem.InfComput.ScL, 38, 563-574. 

Hosoya and Yamaguchi have introduced the sextet polynomial as a book-keeping device for different types of structures that occur in the generalized Clar polynomial. By definition, the constant term of the polynomial is equal to one (and corresponds to the molecular graph of the benzenoid considered). The coefficients of various powers of A that constitute the polynomial count the number of structures having different number of jr-sextets, and the coefficient of the term tells how many k sextets are in the molecule. For example, for benzo[a]pyrene the sextet polynomial is 5(a) = 1 + 5a + 3a, which means that there are three structures with two sextets, five with one sextet, and one structure with no sextet. In Table 49 we have listed sextet polynomials for a collection of smaller benzenoid hydrocarbons, as reported by Hosoya and Yamaguchi.The coefficients of the sextet polynomials have been referred to as the resonant sextet number , that is, the number of ways in which k mutually resonant sextets can be selected in a benzenoid structure. 

Hosoya, H. Clar s Aromatic Sextet and Sextet Polynomial. 153, 255-272 (1990). 

The investigations outlined in this article are grouped into four sections (a) questions concerned with the structure of benzenoid systems, (b) spectral properties and graph polynomials, (c) works related to Kekule and Clar structures and (d) topological indices. It will become clear, however, that all these researches are intimately interrelated and that there exist quite a few unexpected connections between them. 

A more well known result is the case when a homologous set of linear acenes is defined where K(B.) = j+1 In Fig 14 we ow a number of homologous sereis of benzmoids Table 6 lists Clar [27] and sextet [39] polynomials for such equivalence classes ... 

Table 6 Clar and sextet polynomials of equivalent graphs shown in Fig. 14.

Ohkami N, Motoyama A, Yamaguchi T, Hosoya H, Gutman I (1981) Graph—Theoretical Analysis of the Clar s Aromatic Sextet — Mathematical Pro rties of the Set of the Kekule Patterns and the Sextet Polynomial for Polycyclic Aromatic Hydrocarbons. Tetrahedron 37 1113... 

Characteristic polynomial and related, refs 950— 957 graph spectra and related, refs 958—960 automorphism, refs 961—966 enumerations of Kekule structures, refs 967, 968 enumeration of walks and related, ref 969 more on Kekule structures, refs 970— 977 Pauling bond orders, refs 978—980 more on Clar structures, refs 981—990 aromaticity, refs 991—994 and fullerenes, refs 995, 996. 

In the discussion the Clar structures of fullerene Ceo, I neglected to mention two papers in which these structures ° °° have been discussed. S. El-BasiP°° was the first to report on the five Clar structures of buckminsterfullerene, while mathematicians W. C. Shiu and P. C. B. Lam (from Hong Kong) and H. Zhang (from P. R. China) elaborated on the corresponding sextet polynomials of buckminsterfullerene. I am grateful to Professor W. C. Herndon (El Paso, TX), who drew my attention to these recent papers on Clar structures of fullerenes. 

---