Normal Single Coronoids

In Sect. 2.2 the half essentially disconnected (HED) single coronoids are defined (provisionally) as the normal single coronoid which are not regular. A direct definition of the HED single coronoids, compatible with the provisional definition of Sect. 2.2, is furnished through the theorem of the next paragraph. 

Theorem 8.5 A normal single coronoid G is not regular if and only if —... 

Proof of Sufficiency. Suppose that the conditions (1) — (3) are fulfilled for a normal single coronoid G. Then we shall prove by contradiction that G cannot be regular. Assume that the set K of G can be divided into K and K2 such that K (z = 1, 2) contains some fixed single bonds iti an edge cut of type 2, and R, R2 is a standard combination. Suppose... 

In the first place, the Kekulean single coronoids are classified into normal (n) and essentially disconnected (e) systems. Obviously, n + e accounts for all the Kekulean systems. 

It was conjectured that the provisional definition of HED single coronoids as the normal systems which are not regular, always would lead to systems reasonable to be dassified as HED ... 

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

Figures 1 and 2 emphasize the anomalous behaviour of Kekule structure counts for irregular single coronoids. However, it is not always so that the K numbers decrease through similar sets of coronoids as in these figures. The opposite situation is observed in Fig. 3, where again one he and one e system is produced by successive additions of hexagons to a regular (r) coronoid six hexagons are added each time. In this case the K numbers happen to increase udth increasing h in the "normal" way.

Table 2 shows the numbers of single coronoid isomers according to the neo classification (n normal e essentially disconnected o non—Kekulean). Furthermore, the non—Kekulean systems, o, are classified according to their color excess (A). The table is arranged in a way which was found suitable for benzenoid isomers (Brunvoll, Cyvin BN and Cyvin 1992b). All the numbers in this table, although they have not been given before explicitly, can be deduced firom the data of Volume I, Tables 1-9.1 to 1-9.8. 

Fig. 7.2. Forms of single coronoid isomers. Abbreviations 71 normal 0 non—Kekulean, with the A values indicated as subscript - 0. When the associated benzenoid is strictly peri condensed, the contour of its excised internal structure is given by heavy lines.

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). 

Theorem 8.S A single coronoid G is normal if and only if there are two Kekul structures Mi and M2 of G such that C and C" of G are Mi- and M2-alternating cycles, respectively. 

It is evident that the tearing down or building up of regular single coronoids as described above, passes exclusively through normal benzenoids and regular coronoids in every step. 

---