Kekule structure counting

As usual K B is used to denote the Kekule structure count of a (generalized) benzenoid system B. 

The enumeration of Kekule structures for rectangle-shaped benzenoids is treated. Combinatorial formulas for K (the Kekule structure count) are derived by several methods. The oblate rectangles, Rj(m, n), with fixed values of m are treated most extensively and used to exemplify different procedures based on the method of fragmentation (chopping, summation), a fully computerized method (fitting of polynominal coefficients), application of the John-Sachs theorem, and the transfer-matrix method. For Rj(m, n) with fixed values of n the relevant recurrence relations are accounted for, and general explicit combinatorial K formulas are reported. Finally a class of multiple coronoids, the perforated oblate rectangles, is considered in order to exemplify a perfectly explicit combinatorial K formula, an expression for arbitraty values of the parameters m and n. 

The Kekule structure count (or number of Kekule structures) is denoted by K. More specifically, when B is a benzenoid, then K B is used to identify its Kekule structure count. Thus, for instance, K (Rf(3,3) = 64andK Rj(3,3) = 650 cf. Fig. 1. 

A prolate rectangle, R (m, n), is an essentially disconnected benzenoid [1-3]. Hence the Kekule structure counts are easily obtained by... 

The fully computerized method is actually a numerical coefficient fitting for polynomials in general, but was developed in connection with Kekule structure counts. It was used to reproduce the K formula (10) for Rj(4, n) [9]. Furthermore, it allowed to proceed to the m value one unit larger, yielding [10]... 

The famous John-Sachs theorem [26] gives the Kekule structure count of a benzenoid in terms of an np x np determinant, where np is the number of peaks [27], equal to the number of valleys [27]. As pointed out by Gutman and Cyvin [28] the elements of this John-Sachs determinant may be identified with K numbers of certain benzenoids, occasionally degenerated to an acyclic chain (polyene), or zero. An application to the oblate rectangles gave the result [2,13] ... 

Powers of the transfer matrix account for propagation between local states that are more distant. That is, (g T p) gives the number of ways of propagating from p) across / cells to q). To count Kekule structures on Rj(/n, n) we note that there are m — 1 cells and that any one of the local states may occur at the boundaries of the initial and final cells. Thus the total Kekule structure count is... 

A study of the Kekule structure counts for perforated rectangles gave the result ... 

The recursions of the preceding section can be alternatively cast into an especially elegant form for polymer graphs. The Kekule-structure count KL for a polymer chain of length L monomers can [139-144] quite generally be cast into the form of a trace... 

The most general Kekule-structure-count method of the present type was devised by Kasteleyn [146], though there is slightly earlier work for different special cases [33,147]. This too involves certain matrices, most simply the graph adjacency matrices /4(G) with rows columns that are labelled by the sites of G and elements that are all 0 except those Aab=+ with a b adjacent sites in G. Then Kastelyn shows how for "planar" graphs to set up a "signed" version (G) of this matrix with half of its +1 elements replaced by -1 such that... 

Thus many of the Kekule-structure enumeration methodologies of section 4 have been shown to rather neatly extend to conjugated-circuits enumerations, with but modest trouble beyond the overall Kekule-structure count K(G). 

In [tpCBj, ix)/cp(B2, ix)] tends to 2 In [K B1 /K B2 ]. Nevertheless, the assertion [68] that the energy difference of benzenoid isomers is proportional to the difference between the logarithms of their Kekule structure counts was not confirmed by later investigations (see Sect. 7.2). 

The results of enumerations and classifications of polyhexes are reviewed and supplemented with new data. The numbers are collected in comprehensive tables and supplied with a thorough documentation from an extensive literature search. Numerous forms of the polyhexes are displayed, either as dualists or black silhouettes on the background of a hexagonal lattice. In the latter case, the Kekule structure counts for Kekulean systems are indicated. Emphasis is laid on the benzenoid systems (planar simply connected polyhexes). 

Another important quantity for a benzenoid is the Kekule structure count or K number. A Kekule structure, being a typical concept from (mathematical) chemistry, corresponds to a 1-factor or perfect matching in mathematics. 

In Fig. 18 the forms of the Dik catacondensed benzenoids up to h = 25 are displayed. They have been given previously [79], Being catacondensed, all these systems are normal and therefore Kekulean. The Kekule structure counts (K) are given in the figure. 

A substantial amount of additional enumeration data for normal benzenoids and some data for essentially disconnected benzenoids are available, but shall not be reproduced here. They were produced in the course of the extensive studies of the distribution of K, the Kekule structure count. 

Much information can be extracted from Knop et al. [5], where all benzenoids with h < 9 are depicted. These computer-generated pictures are ordered according to the numbers of internal vertices (n,) within each h value. The Kekule structure counts are indicated (K > 0 for Kekulean and K = 0 for non-Kekulean systems). In Tables 2 and 3 this reference is quoted in appropriate places for some total Kekulean and total non-Kekulean systems. We have not taken into account the corresponding mammoth listing for h = 10, on which it was informed by Knop et al. [44]. It was stated that a very limited number of copies were available for distribution in 1984. We are not in the possession of any of these copies. 

Balaban, A.T., Liu, X., Cyvin, S.J. and Klein, D.J. (1993b). Benzenoids with Maximum Kekule Structure Counts for Given Numbers of Hexagons. J.Chem.Inf.Comput.Sci, 33,429-436. 

Cash, G.G. (1998). A Simple Means of Computing the Kekule Structure Count for Toroidal Polyhex Fullerenes. J.Chem.lnf.Comput.Sci., 38, 58-61. 

Mishra, R.K. and Patra, S.M. (1998). Numerical Determination of the Kekule Structure Count of Some Symmetrical Polycyclic Aromatic Hydrocarbons and their Relationship with n-Electron-ic Energy (A Computational Approach). J.Chem.Inf.Comput.Scl, 38,113-124. 

Many physico-chemical properties and biological activities seem to fall within the domain of additive properties. Examples of constantive properties are local molecular properties, such as dissociation energy for a localized bond or ionization potential. Characteristic multiplicative properties are wave functions, Kekule structure counts, and probabilities. The derivative properties are associated with the corresponding multiplicative property T. 

Euleric corollary of Section 9.4.1) have been identified, with a crucial part of the completeness of the construction going back to some mathematical work. " The possible topological equivalence classes of embeddings are a further problem but even all of these may have been found. The problem of Kekule-structure counts was first done for a few cages, then for a few classes of tori, " and is now completely solved. Moreover, a general analytic treatment of the Hiickel eigenspectrum has been made. ... 

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