Circular benzenoid

Circular Benzenoids and One-Isomer Benzenoid Series 8.1 Circular Benzenoids... 

A circular benzenoid [8] is defined by having the largest number of hexagons (Amax) for a given perimeter length or s value. It has been found [15,47]... 

Fig. 10. The smallest circular benzenoids with the non-benzenoid C7H7 included...

Each formula for a circular benzenoid in Fig. 9 is marked by a circle (with some other symbol inside it). It can be shown that, apart from C6H6 and the appropriate formulas in the two first (abnormally high) steps in the periodic table, the circular benzenoids always have their formulas at the top of three-formula steps. 

There is exactly one circular benzenoid for each formula of O. Furthermore, these benzenoids exhibit six characteristic shapes to be accounted for in the following. These shapes were at least known to Balaban [48], who investigated their important role in the studies of annulenes cf. also some later works in this connection [15,49]. 

Figure 10 shows the forms of the smallest circular benzenoids. For the sake of systematization the non-benzenoid system for C7H7 is included. All larger systems are obtained by circumscribing those which are displayed. In Fig. 10, the second row (k = 1) is obtained by circumscribing the systems in the first row (k = 0). The different shapes (in the columns of Fig. 10) are identified by the labels s = 0, 1, 2, 3, 4, 5. 

Proposition 6 Amy circular benzenoid with h = 7 and h > 10 has all its free edges in the six Z.3-mode hexagons which it possesses. 

With the aid of the previous analyses of annulenes [15, 48, 49] the following alternative expression was found for the formulas of the circular benzenoids in general. 

In Table 2 the formulas of quite a few of the first (smallest) circular benzenoids are listed. 

In this section, two formulas (n s) for O, the circular benzenoids, are reported, viz. (a) and (b). In (a) an elaborate floor function (17) is present. It is avoided in (b) the floor function therein assumes only the value 1 for s = 0 and vanishes for > 0. But instead, the form (b) contains two parameters (k, e), while (a) contains only one, viz. s. We shall refer to the two formulas for O as being in (a) the Harary-Harborth picture and (b) the Balaban picture, respectively. It is recalled that they originate from the analysis of Harary and Harborth [44] in the case of (a) and from Balaban [48] in the case of (b). Under this scope all the inequalities and deductions from them in the previous chapter [8] are in the Harary-Harborth picture. 

In order to inspect the first term on the right-hand side of Eq. (27) closer we ask ourselves first on which circular benzenoid the two-contact addition according to (a) should be executed. It is clear that the formula of this circular benzenoid is obtained by subtracting C3H from C HS. In terms of the three-parameter code it was found that ... 

When we have identified the circular benzenoid o, 3, /, which should be... 

Another interpretation of the location of the G° formula relates it to the circular benzenoid formula just above the one for j — 1,5. If O is the circular benzenoid with the formula given by (29), then its excised internal structure has the formula... 

A circular (single) coronoid (Cyvin SJ 1991b) may be defined in analogy with a circular benzenoid (Cyvin 1992c Brunvoll, Cyvin BN and Cyvin 1992b Cyvin SJ, Cyvin and Brunvoll 1993e). 

The circular single coronoids are represented by the dots on the stippled curve in Fig. 5. It is an important feature that the circular coronoids form a subclass of the extremal coronoids (of the same genus). Consequently, the circular coronoids are naphthalenic. Furthermore, a circular single coronoid is a circular benzenoid perforated by a naphthalene hole. 

A circular single coronoid, 0, (Par. 4.6.2) is a circular benzenoid perforated by a naphthalene hole (cf. also Par 5.5.3). Cyvin SJ (1991b) presented a complete mathematical solution in terms of combinatorial formulas for the numbers of 0 isomers. 

The enumeration of O isomers reduces clearly to the counting of distinctly different positions into which a naphthalene hole can be placed within a circular benzenoid. The position of a naphthalene hole is uniquely determined by the central edge of naphthalene. Hence the process ends up with counting of edges. 

Assume that 0 N S) is a circular benzenoid which can be perforated by a phenalene hole to produce a single coronoid 0(n s). Then one has for the formula coefficients l T=n+l, 5=s — 3. All the isomers of the class 0 were enumerated. This is not a complete determination of the cardinalities because there always exist at least naphthalenic single coronoids with the... 

The six characteristic shapes of circular benzenoids are recognized as in the case of circular single coronoids (Fig. 4.6), and are again indicated by the parameter e = 0,1, 2, 3, 4, 5. The smallest of the systems, which presently are associated with k = 0, are displayed in Fig. 4 three coronoids and three degenerate coronoids. All the C H formulas under consideration are given (in the Harary-Harborth picture) by... 

Fig. 7.4. The smallest circular benzenoids perforated by a phenalene hole, augmented by three degenerate coronoids (in the top row).

Table 7.8. Formulas for circular benzenoids perforated by one phenalene hole each. ...

Table 7.10. Numbers I of isomers of circular benzenoids perforated by one phena each. For the first formulas k < 6), see Table 7.8. ...

Exactly all the circular benzenoids 0 N S) with 5 > 18 can be perforated by a coronene hole to produce single coronoids 0(n 5). Then one has i T=n+6, 5=5 — 6. A detailed treatment on a complete enumeration of the isomers of the class 0 has been published (Cyvin BN, Brunvoll, Chen and Cyvin 1993). The smallest system of this class is kekulene. In the cited reference a coronoid with coronene hole(s) was referred to as "coronenic . We quote a nice sentence therefrom "A coronenic coronoid has a coronene corona hole. We shall not use the term "coronenic in the following, although it is analogous with our term "naphthalenic (Definition 3.3). A brief account on the title systems and their enumeration is given in the next two paragraphs. 

Fig. 7.5. The smallest circular benzenoids perforated by a coronene hole.

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