Extremal coronoid

An extremal coronoid (single or multiple) is defined in the same way as an extremal benzenoid (Harary and Harborth 1976 Cyvin SJ 1992c BrunvoU, Cyvin BN and Cyvin 1992b). 

Definition S.4 An extremal p-polyhex, where g is fixed, is defined by having the maximum number of internal vertices (n ) for a given number of hexagons (h) = ( j)uiax( )- 

An extremal tuple coronoid shall frequently be identified by the symbol A in the following. A coronoid which is not extremal, is sometimes referred to as a nonextremal coronoid. 

A formula for the upper bound of n in polyhexes has been launched by Cyvin SJ and BrunvoU (1990) as 

The special case of this equation for = 1 is given in Vol. I - 3.2.3 and elsewhere (Cyvin SJ and BrunvoU 1989). The general form of (42) was found very simply by inserting if and from eqn. (39) into the well known inequality for benzenoids g = 0), viz. (Harary and Harborth 1976 Gutman 1982) 

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Below we give a useful definition in connection with the above discussion. Consider a tuple extremal coronoid and denote its associated benzenoid by A. 

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

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

Some introductory observations about extremal coronoids are reported below. They are foUowed by comments which tend to justify the inherent assumptions. However, the observations are not to be considered as rigorously proved. 

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. 

Consider two extremal coronoids AJ h) and A (/i), along with their associated benzenoids, say B and B, respectively. Here B may be identical with B, namely if it is possible to perforate B = B by naphthalene holes in two different ways so as to create the two nonisomorphic systems A (h) and A h), In general B and B must have the same invariants as indicated by B(H,iNT), B (H,A ). This is a consequence of Observation 1. It foUows that B and B also have the same formula, say C yH, and the same perimeter length, say N. The connections... 

A simple deduction from Observation 1 and the above discussion is given below. Consider two extremal coronoids A h) and A h). 

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) ... 

It was stated (Par. 3.6.2) that n - = 0 is realized in a y-polyhex for every g and h. It should also be clear that, if a catacondensed y-tuple extremal coronoid exists, then it can only occur for h = h g). It is always possible, namely, to add a hexagon to a catacondensed coronoid so that it becomes pericondensed. (The same property is also found for benzenoids with one exception benzene. Accordingly, both benzene H = 1) and naphthalene H = 2) are catacondensed extremal benzenoids.) The above discussion, along with the last lines of Par. 3.5.5, lead to the following conjecture. 

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. 

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. 

A single coronoid which has h = for a given s is clearly an extremal coronoid, A. However, the property h = cannot serve as a definition for extremal single coronoids. As... 

The circular single coronoids are represented by the dots on the stippled curve in Fig. 5. It is an important feature that the circular coronoids form a subclass of the extremal coronoids (of the same genus). Consequently, the circular coronoids are naphthalenic. Furthermore, a circular single coronoid is a circular benzenoid perforated by a naphthalene hole. 

An extreme coronoid (not to be confused with an extremal coronoid) is analogous with an extreme benzenoid (with a few exceptions specified below) cf. Cyvin SJ (1992c), which has also been called an extreme—left benzenoid (Cyvin SJ 1992c Cyvin SJ, Cyvin and Brunvoll 1993e). 

The focussing on formulas for single coronoids is continued in this chapter. It starts with a treatment of sequences of associated formulas for extremal single coronoids. This study leads to another subdivision of the extremal coronoids ground forms and higher members. The pertinent sections can be considered as a preparation to the enumeration of coronoid isomers, which is treated in the subsequent chapter. Also the process of building-up (Sect. 6.5) is highly relevant to the enumeration of isomers. 

The actual forms of a number of single coronoid isomers are shown in Fig. 2. All these forms for < 14 are found to be consistent with the relevant numbers and classifications of Table 2 and of Table 4. The depictions go somewhat beyond h = 14. All the depicted forms for h> IS are extremal coronoids. 

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