Degenerate single coronoid

Definition 2.2 A special degenerate single coronoid is a d enerate single coronoid where the edge(s) not belonging to hexagons are common to the outer boundary and the hole. 

In I—Fig. 2.4 the three systems of the upper row are not special d enerate coronoids, but this is the case for the bottom system. Here are four additional examples of degenerate single coronoids ... 

Two of them ( 2 22 26 22 special degenerate single coronoids, while the other... 

In the case of special degenerate single coronoids, as well as in single (nondegenerate) coronoids, one speaks about the outer and inner perimeter, viz. C and C", respectively, but now some edge(s) belong to both C fl C" 0. 

Definition 2.S A generalized single coronoid is either a single coronoid or a special degenerate single coronoid. 

If one wants to avoid the term "special degenerate single coronoid", then one has the synonymous designation "degenerate generalized single coronoid" at hand. 

The classifications normal and essentially disconnected are immediately applicable to degenerate single coronoids. 

In the first case (i) of the above diagram, the effective units are four benzenoids (two pyrenes and two benzenes) in case (ii) the effective units are two benzenoids (anthracenes) and one coronoid (kekulene) in case (iii) the effective units are two benzenoids (naphthalenes) and one special degenerate single coronoid (condensed [18]annulene, viz. hexabenzo[6cdc/,fcim7io][18]-annulene). 

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.

The system c(C), however, is not always a coronoid. As an example, Fig. 1 shows the exdsed (double) coronoid from Fig. 3.1 it is seen to be a degenerate coronoid system. Figure 2 shows the exdsed polyhexes from Fig. 4.2, viz. five benzenoids and four single coronoids, the latter systems belonging to the proper topic of the present chapter. The examples are supplemented by... 

Some explanations should be attached to this somewhat complicated definition. It is dear that a drcumscribed extremal (single) coronoid is circumextremal write circum—A = P. Then it is also dear that excis-P = A. Assume now that the naphthalene hole of P is moved as near as possible to the outer perimeter, and call the new coronoid P. Then P and P are isomers, but P cannot be excised to give another coronoid excis—P becomes a degenerate system. Nevertheless, P falls under the definition of drcumextremal coronoids. An illustration is given below. 

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 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... 

Assume now that G is a degenerate generalized single coronoid, and A is a set of nonadjacent vertices, all of them on the outer or on the inner perimeter of G, but not on the intersection of the two perimeters. 

Theorem 8.S An essentially disconnected single coronoid G has at least two effective units. One of them is a normal benzenoid and an other a normal generalized single coronoid (degenerate or non—degenerate). 

---