Perfect extremal coronoid

Definition S.5 A is a perfect extremal coronoid if and only if A is an extremal benzenoid. 

In connection with Observation 1, suppose that a coronoid C has a corona hole larger than naphthalene. Then we can imagine that a closer packing of the hexagons of C is possible by a partial filling of the corona hole so that the total number of internal vertices increases. In Observation 2 the crucial term ("perfect extremal coronoid") conforms with Definition 3.5 of Par. 3.3.4. It is reasonable to imagine that there is a critical smallest size for an extremal benzenoid, say A, so that A can be perforated with g naphthalene holes, which is taken to be the necessary condition for creating a perfect extremal tuple coronoid. 

CaroBary If is a perfect extremal coronoid, then any extremal coronoid is also... 

In benzenoids and y-tuple perfect extremal coronoids the number of hexagons is equal to the right-hand side of (61) for these cases we write in analogy with eqn. (58) ... 

Pig. 3.4. The perfect extremal coronoids for some of the smallest h values when 1 < < 5. 

Furthermore, one should observe the termination of each row in the right-hand direction. This termination results in the staircase boundary. It is determined by the formulas situated at the extreme—right of each row, namely the formulas which pertain to the extremal polyhexes, as should be dear from the definition of these systems (Par. 3.3.4). When perfect extremal p-tuple coronoids are involved, then the corresponding staircase boundary reflects a part of the staircase boundary for benzenoids. It is specifically the part which corresponds to the benzenoids associated with the tuple coronoids in question. Thus, for instance, the staircase boundary for single coronoids has the same shape as the one for benzenoids when starting from C32H14 ovalene (cf. Table 5, where the start of this staircase boundary for benzenoids is indicated by thin lines). A staircase boundary of this kind, determined by perfect extremal coronoids or extremal benzenoids, shall be referred to as a perfect staircase (boundary). If imperfect extremal coronoids are involved we shall call it an imperfect staircase. 

A perfect extremal coronoid or an extremal benzenoid is characterized by h, n ) in the notation of eqns. (58) and (62). Introduce... 

In the above diagrams the staircase boundaries are indicated by heavy lines. For the portions where they are imperfect they are augmented by stippled lines to the shapes which conform with perfect boundaries. Hence it should be understood that the parenthesized formulas actually do not exist for the coronoids in question. The heavy formulas pertain to perfect extremal coronoids. 

An extremal coronoid which is not perfect, shall be referred to as an imperfect extremal coronoid. 

Moreover, it was found that the catacondensed extremal tuple coronoids for p = 1 and 2 are perfect, while those for > 4 are all supposed to be imperfect. Therefore, in a sense, the unique case of y = 3, for which there is no catacondensed extremal coronoid (cf. Par. 3.5.5), marks a borderline between the perfect and imperfect catacondensed extremal coronoids. 

The algorithm stops when a benzenoid A(h+2y, n +10y) is perforable for the first time, i.e. for the smallest h = oi h= h Then A (/i,n ) represents the smallest perfect extremal tuple coronoid(s) for the chosen g value. 

C83023(0=31, 0=43) represents 20 isomers (Cyvin SJ, BrunvoU and Cyvin 1991d), from which one perforable system was detected and gave a unique C8303i( 23, n 3) perfect extremal quadruple coronoid (see Fig. 4). 

In Fig. 4 the smallest perfect extremal tuple coronoids for = 1, 2, 3, 4 and 5 are shown they occur for h = 8, 13, 19, 23 and 28, respectively. The analysis was pursued, by perforating the appropriate extremal benzenoids, in order to generate the systems with a few higher h values for each g the results are included in Fig. 4. 

It has been mentioned (cf. e.g., Par. 4.3.1) that an A coronoid is naphthalenic and perfect extremal it is obtained from an A benzenoid by perforating it vdth a naphthalene hole. 

In Table 3 "everything is perfect". The formulas at the extreme right of each row pertain to perfect extremal single coronoids, and therefore the right-hand boundary is a perfect staircase cf. Par. 3.7.3. 

Already in the original work on extremal uple coronoids (Cyvin SJ and BrunvoU 1990) it was pointed out that eqn. (42) is not "perfect" in the sense that the upper bound as given therein is not realized for all possible pairs of (h, g) values. It was verified, however, that this upper bound is realized for all the possible values of h when y = 1 and y = 2, in addition to y = 0. But in general one has to write... 

In this chapter the findings for invariants of polyhexes (cf. Sect. 3.2) are summarized for the special case of single coronoids ( = 1) and extended in a more detailed treatment. It comes as a supplement to the treatment of invariants of single coronoids in Vol.I—3.2.2. In particular, the upper and lower bounds of several invariants are treated extensively. This leads naturally to a treatment of extremal single coronoids (cf. Par. 3.3.4), which invariably are perfect. 

Figure 3 shows a dot diagram for the formulas of single coronoids as in Fig. 5.8. However, more detailed information about the extremal (A) systems is included formulas for ground forms and higher members, both perfect and imperfect, are indicated.

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