Regular Single Coronoids

When a normal benzenoid B with h hexagons is subjected to a normal addition, then the Kekule structure count of the new normal benzenoid with h+l hexagons is larger than the Kekule structure count of B (Cyvin SJ and Gutman 1986 Cyvin SJ, BrunvoU and Cyvin 1989b). In a normal addition, the added hexagon acquires by definition the mode or L. More 

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It remains to define the regular single coronoids. In order to give an idea of the approach which was adopted (precise definitions are saved for a subsequent chapter), consider the regular coronoid ... 

The purpose of this diagram is twofold (1) It should recall the notation of modes of hexagons cf. Vol. 1-3.2.1, especially I-Fig. 3.1, and elsewhere (Cyvin SJ and Gutman 1988 Gutman and Cyvin 1989). (2) It should illustrate the definition of a regular single coronoid, which implies... 

A definition of regular single coronoids is illustrated in Sect. 2.2. Here we give a precise formulation. 

It follows that any regular single coronoid can be built up by regular additions starting from benzene, whereby a corona-condensation should be executed only once. 

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. 

Proof of Necessity. Suppose that G is a regular single coronoid. By definition, G can be obtained from benzene by regular additions in three steps (cf. Definition 8.7) ... 

Notice also that the conditions (2) and (3) are indispensable. A regular single coronoid may also have two subsets of Kekule structures, which together constitute the complete set of Kekule structures, and where each subset possesses fixed single bonds. 

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 ... 

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.

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.

Here h = (2fl+l)6 for a > 1. Unity must be added for a = 1 in order to comply with the benzenoid drcumcoronene h = 19). All the other systems (a > 1) are regular pericondensed single coronoids. The result for the Kekule structure counts reads (Cyvin BN, BrunvoU and Cyvin 1988) ... 

Definition 8,7 A single coronoid is regular if it can be subjected to a regular tearing down, hexagon by hexagon, down to benzene. 

Theorem 8.4 A single coronoid G is regular if and only if there is a Kekule structure M of G such that both C and C of G are M-altemating cycles. 

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... 

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