Associated benzenoid

Associated Benzenoid and Perforated Benzenoid Let denote a tuple coronoid. 

Definition S.l The associated benzenoid to is defined by the benzenoid which emerges when the g corona holes of are completely filled by hexagons. 

It is noted that the associated benzenoid is determined unambiguously when is given. 

Assume that C is a naphthalenic tuple coronoid with (h, n and the formula Furthermore, let the associated benzenoid to C be denoted by B, of which some selected... 

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

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... 

In this case the benzenoid associated with A (h) is extremal. Let it be denoted by A jHT), where H is its number of hexagons. Since the associated benzenoid to A (h), say A has the same invariants as A(if) it must also be extremal and can be denoted by A ( in order to indicate that it has the same number of hexagons as A H). 

For 17. This corresponds to the formula C64H26 and the associated benzenoids... 

Here it is found by inspection that three naphthalene holes cannot be introduced in any of these benzenoids. Hence it is inferred that no triple coronoid with the formula C64H26 can be constructed. We must therefore look for the possibilities with h > h. Then the first case to be inspected is h = 18, corresponding to C68H28 and the associated benzenoids C68H22( =24, =30). These benzenoids are nonextremal and count 789 isomers (Stojmenovic et al. 1986 BrunvoU and Cyvin SJ 1990), a too large number to be inspected by sight, even if the depictions were available. However, two of these isomers perforated by three naphthalene holes, have actually been constructed (see Fig. 2) as the smallest catacondensed triple coronoids, in consistency with the present analysis. 

For g = Aj h = 21. One obtains as a starting point the formula C78H30, which corresponds to the associated benzenoids C78H22( =29, JV=40). The actual four isomers (Cyvin SJ, BrunvoU and Cyvin 1991d) of these extremal benzenoids were inspected, but none of them could be perforated by four naphthalene holes. Consequently, we turn to h = 22, the formula C82H32 and associated benzenoids C82H24(f7=30, —40). There are 1799 isomers of these nonextremal benzenoids. In accord with Fig. 2 it is claimed that only one of them can be perforated by four naphthalene holes in a unique constellation. 

A coronoid is said to be circumscribed if its associated benzenoid is circumscribed. Hence the following definition may be formulated (cf. Cyvin SJ, Brunvoll and Cyvin 1991b Brunvoll, Cyvin BN and Cyvin 1993b also John 1990). 

Definition 5.7 A single coronoid (P) is circumextremal if its associated benzenoid, P, is a drcumextremal benzenoid, and exds-P is the assodated benzenoid to a (smaller) extremal single coronoid. 

The associated benzenoid to G is a ground form benzenoid, while the associated... 

Table 6.1. Fonnula for the first (smallest) ground forms of single coronoids (G) and their associated benzenoids (A) ...

More elaborate definitions or characterizations of the systems in Definition 6.4 could be based on the corresponding associated benzenoids in the style of Definition 5.7 cf. also the explanations below the latter definition (Par. 5.5.2). 

The coefficients of N 5) were transferred to those of a naphthalenic single coronoid, say B(h, n s, whose associate benzenoid is B N 5). Then the expressions (14) read ... 

From the information on B, the associated benzenoid to B, we deduce the following properties of the naphthalenic coronoid B( i s). 

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.

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