Multiple coronoid

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. 

Consider a perforated oblate rectangle as depicted in Fig. 9. Like a rectangle it is a system with two parameters (m, n). A perforated rectangle, Q(m, ), belongs to the multiple coronoids it has m — 1 corona holes. 

The term polyhex has been used about benzenoids and coronoids together [18]. Benzenoids are polyhexes without holes, while a single or multiple coronoid is a polyhex with one or more holes, respectively. There is also a tendency to use the term polyhex in a more general sense [16]. Here we shall use it as a universal term for all hexagonal systems which are to be considered. 

Any (poly)circulene, which may be a (multiple) coronoid, is a multiply connected polyhex. 

Only benzenoid systems are considered. According to the adopted definition they are the planar, simply connected polyhexes. Consequently all circulenes (including coronoids, multiple coronoids and polycirculenes) are excluded, and all helicenic systems (helicenes, helicirculenes) are also excluded. 

Fig. 1.2. Multiple coronoids with symmetry ("laceflowers") and h < 54, where h designates the number of hexagons. 

Multiple coronoids (systems with more than one hole each) of symmetry have been called laceflowers (Cyvin SJ, Brunvoll and Cyvin 1989a Cyvin BN, Brunvoll, Chen and Cyvin 1993). The two cited references contain depictions of these pretty systems up to 54 hexagons they are reproduced in Fig. 2. Also the aesthetic qualities of fractal benzenoids (Klein, Cravey and Hite 1991) can hardly be denied. A representative is shown in Fig. 3. The particular set to which it belongs, starts with the two benzenoids CgHg (benzene) and 2 12 Then... 

Benzenoids and single coronoids represent two main classes of polyhexes (Sect. 2.1). The inclusion of multiple coronoids represents an extension in one direction. Classes of multiple coronoids are defined and discussed in the next chapter. Here we shall concentrate upon polyhexes without holes (as benzenoids) and those dth one hole each (as single coronoids) and discuss an extension which involves geometrically nonplanar (helicenic) systems. We shall find it expedient to define a class where the members are restricted to helicenic systems which can be represented by planar graphs. In the next section, polyhex systems which cannot be represented by planar graphs, are exemplified. 

Table 3.1. Invariants of polyhexes benzenoids (p = 0), single coronoids g = 1) and multiple coronoids g > 1). ...

The designation "perforated rectangles" for certain single and multiple coronoids, of which the Kekul6 structures have been studied (Cyvin SJ, Cyvin and BrunvoU 1989a Chen and Cyvin 1989 Cyvin SJ, Cyvin, BrunvoU and Chen 1989 Chen, Cyvin et al. 1990), is compatible with the above definition of perforated benzenoids. 

Coronoids with exclusively naphthalene holes play important roles in different contexts, as we shaU see in the foUowing. This is true for both single and multiple coronoids. 

The interesting and important question about the smaUest multiple coronoids is stiU an open problem. The tentative solution presented below is not supported by rigorous proofs. A precise formulation of the problem is Find the minimum number of hexagons for a polyhex when g is given. In other words, determine the function From the systematic... 

Assumption A smaUest multiple coronoid can be found as a catacondensed naphthalenic system, where the naphthalene holes are paraUel. 

The three systems depicted above are also consistent with the assumptions formulated in this paragraph. Herefrom it is clear that these assumptions are not sufficient for a multiple coronoid to be a smallest system. It is even not implied that the formulated assumptions are necessary. 

In the second example of Par. 3.5.3 the unique smallest multiple coronoid for = 7 was derived from coronene according to the algorithm under consideration. Another such system emerges from C25H13 benzo[/u]anthanthrene. This nonextremal benzenoid has 25 internal vertices (iV) against = 24 (cf. Table 3). On the other hand, its minimum number of parallel... 

Table 6 shows the numbers of isomers for some classes of multiple coronoids, most of them taken from the enumerations of Cyvin SJ and BrunvoU (1990). The supplementary values for > 2 in Table 6 are present results, which can be taken out from Figs. 2 and 4 in addition, it is inferred from Fig. 2 CnoH4o g = IC162H54I = C2ooH64l2 3 = 1, Ci24H44 y = Ci38H48 g... 

Excising. Any single or multiple coronoid C (as well as any benzenoid) can be excised. 

Cyvin SJ, Cyvin BN, BrunvoU J (1989a) Kekule Structure Counts and Multiple Coronoid Hydrocarbons. Chem Phys Letters 156 595... 

---