Sextet polynomial

Various algebraic and combinatorial aspects of Claris aromatic sextet theory are outlined in the recent book [3] and the recent reviews [50, 51, 89]. Therefore, in this paragraph we will just point out a few details related to the author s own research and mention the most recent developments in the filed. 

The crucial impetus for these investigations was given by the short paper of Hosoya and Yamaguchi [90] in which they introduced the numbers s(B, k) and the sextet polynomial 

The numbers s(B, k) count the generalized Clar structures of the benzenoid system B, containing exactly k aromatic sextets. For instance, for B = benzoan-thracene we have (see Fig. 2)  

This example is also an illustration of the peculiar fact that the number of generalized Clar structures coincides with the number of Kekule structures, i.e. 

The identity (7) is just a consequence of a one-to-one correspondence between generalized Clar and Kekule structures. In particular, the i-th Clar structure in 

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

Fig. 1 One-to-one correspondence between the Kekuie and sextet patterns of benzanthracene to give its sextet polynomial...

In Table 2 are given the sextet polynomials for the lower members of aromatic hydrocarbons together with their K(G) numbers. The sextet polynomials for larger benzenoid graphs are extensively tabulated and discussed [13, 14]. As already mentioned, for a thin benzenoid or coronoid system there is exactly a one-to-one relation between the Kekuie patterns and sextet patterns. In other words the following equality is obeyed,... 

Table 2, Sextet polynomials of lower members of benzenoid hydrocarbons.

Although the variable x in the sextet polynomial Ba(x) does not mean anything other than what holds the power k and the coefficient r(G, k), Ba(x) can be formally differentiated with respect to x as... 

Fig. 2. Twenty Kekule and sextet patterns of coronene to give its sextet polynomial, Bg(x) = 1 + 8x + 9x2 + 2x3. The pattern 3 is the super-sextet pattern. See Table 2 and Fig. 4...

Randic [21] and Aihara [22] independently proposed an idea of the index of local aromaticity, ILA, and overall index of aromaticity, OIA, based on the counting of the Kekule patterns. However, these concepts were found to be closely related to the sextet polynomial and its derivative as [9]... 

Theorem 12 provides a straightforward and general method for the calculation of the sextet polynomial [97, 98]. 

Corollary 15.3. If B is an unbranched catacondensed benzenoid system then all the zeros of its sextet polynomial are real and negative numbers. 

A further method for calculating the sextet polynomial of an unbranched catacondensed benzenoid molecule, not based on the Gutman-tree concept, was reported in [113]. 

We have already examined a few graph invariants which, according to the above definition, could be included among the topological indices of benzenoid molecules. These are the number of Kekule structures, the eigenvalues, spectral moments, (coefficients of) the characteristic and sextet polynomials, to mention just some. The total 7t-electron energy is surveyed elsewhere in this volume. 

Hosoya and Yamaguchi [99] published the sextet polynomials systematically for all Kekulean benzenoids with h < 5. This material was supplemented up to h = 6 by Ohkami and Hosoya [100]. 

Table 4 Partition sequence (above) and sextet polynomial (below) for the zigzag benzenoids-...

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

B, 1) is the value of the sextet polynomial, o (B x) for a benzenoid system B when xsl Sextet polynomials were first introduced in H. Hosoya and T Yamaguchi, Tetrahedron Letters, 52, 4659 (1975). 

Claris theory of the aromatic sextet can be formulated in terms of the coefficients of the sextet polynomial. 

Theorem. (Gutman) Every sextet polynomial is an independence polynoaual. 

Readers interested in rook polynomials should consult references [75] and [76]. The sextet polynomial will be introduced later on. 

G.4.3. Defikition. (Hosoya and Yamagadu [142]) The sextet polynomial ofGis defined as ... 

The theory of the sextet polynomial is cuTFently rapidly developing [139,142-151,173-178]. One of the moet exciting discoveries in this field was made in [142] and proved in [146]. 

Another graph—theoretical polynomial is the sextet polynomial see, e.g. Ohkami et al. (1981) and references cited therein Ohkami 1990. For kekulene it reads (Ohkami et al. 1981 Zhang, Cyvin and Cyvin 1990) ... 

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