Circular single coronoid

Mcaimum Number of Hexagons and Circular Single Coronoids... 

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. 

The spiral walk (cf. Par. 4.3.3, and in particular Fig. 4) generates exactly one circular single coronoid for every 5 value. Specifically, the first circular single coronoids are generated by... 

After the above discussion it is clear that the generation of circular single coronoids for h... 

Here we give the first detailed analysis of the circular single coronoids, 0 (but see also the work by Cyvin SJ 1991b). 

The depictions of circular single coronoids in Fig. 6 are augmented by two degenerate coronoids, which fit into the scheme when attributed to k = 0 for = 0 and 1. In fact, the scheme can be extended further by the following degenerate coronoids (starting from [lOjannulene). 

Fig. 4.6. The smallest circular single coronoids, augmented by two degenerate coronoids. The inscribed numerals indicate the number of last added hexagon during the spiral walk.

Let a circular single coronoid be identified by 0(n° 5°). In the following the expressions for the pertinent formula coefficients are deduced in both "pictures". 

The generating function (76) reproduces the formulas of circular single coronoids (cf. Table 2) as is indicated in (70). The degenerate systems (formulas for s < 16) are excluded. 

The circular single coronoids form a subclass of the extremal single coronoids (cf. Par. 

Let 0(nP s ) denote a circular single coronoid as in Chapt. 5. Furthermore, introduce... 

All extremal single coronoids A are naphthalenic. The circular single coronoids 0 are also extremal they form a subclass of the A systems. Some expressions for circumscribing naphthalenic single coronoids, C, are given below. 

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. 

Fig. 7.3. Forms of some circular single coronoid isomers.

It is recalled that the circular single coronoid isomers are defined in terms of two... 

Table 7.6. Numbers of isomers of circular single coronoids explicit formulas / generating functions I x).

Listing of Algebraic Results. Table 6 contains the final results for the total numbers I of nonisomorphic isomers of all circular single coronoids f, k as functions of and the corresponding generating functions I x). Some details, especially for the numbers of isomers of specific symmetries, are given in the following. 

Numerical Results. The numerical values of I for all circular single coronoids e. A with k < 20 are listed in Table 7. 

Table 7.7. Numbers /of e, k isomers of circular single coronoids. For the first C H...

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... 

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