Essentially Disconnected Single Coronoids

The terms junction and effective unit , which were introduced for essentially disconnected benzenoids (BrunvoU, Cyvin BN, C rvin and Gutman 1988), are transferable to coronoids (cf. Vol. 1-3.3.4). 

Definitions 2.4- A junction of an essentiaUy disconnected coronoid C is the set of hexagons which possess fixed bonds. The effective units of C are obtained on deleting all edges which are fixed single bonds, and aU fixed double bonds together with their end vertices. 

In the bdow diagram, three essentially disconnected single coronoids and their effective units are depicted. The junctions are painted black. 

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). 

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Whereas the scheme in Sect. 2.1 applies to both benzenoids and single coronoids, a finer classification for the Kekulean single coronoids was found to be needed during the studies of Kekul6 structure counts for these systems. In this way the concept half essentially disconnected (single) coronoids was introduced by Cyvin SJ, Cyvin and Brunvoll (1987). This concept is explained in the book of Gutman and Cyvin (1989) and in Vol. 1-3.3. 

It was also confirmed that the Kekule structure count is the same for the two isospectral systems under consideration K = 12636. It is interesting that the pair of systems consists of one half essentially disconnected (HED) and one essentially disconnected single coronoid. The two schemes of Kekule structures for the HED system (left-hand drawing in Chart 4) are indicated below. 

Property 8.2. An essentially disconnected single coronoid G has a fixed single bond. 

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). 

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. 

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

Fig. 2.1. One regular (r), one half essentially disconnected (he), and one essentially disconnected (c) single coronoid. Numbers K of Kekule structures are indicated.

On the other hand, if a Kekulean single coronoid with isolated internal vertices has two black and two white internal vertices in succession around the corona hole, then it is essentially disconnected. This feature is illustrated for the e systems from Figs. 1 and 3 in the right-hand column of the above diagram. 

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. 

In other words, fixed double bonds are always accompanied by fixed single bonds. From this viewpoint we may say that fixed single bonds are the fiindamental feature of essentially disconnected systems. The simple reasoning of the above proof is applicable to other essentially disconnected systems than single coronoids as well. It has been applied to benzenoids by Chen, Cyvin SJ and Cyvin (1990). 

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. 

Definition 8.8 A normal single coronoid G is half essentially disconnected (HED) if and only if —... 

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