Reproducible coronoid

Definition 5.3 A coronoid C is called a reproducible coronoid if and only if C can be circumscribed an unlimited number of times. 

Multiple coronoids (systems with more than one hole each) of symmetry have been called laceflowers (Cyvin SJ, Brunvoll and Cyvin 1989a Cyvin BN, Brunvoll, Chen and Cyvin 1993). The two cited references contain depictions of these pretty systems up to 54 hexagons they are reproduced in Fig. 2. Also the aesthetic qualities of fractal benzenoids (Klein, Cravey and Hite 1991) can hardly be denied. A representative is shown in Fig. 3. The particular set to which it belongs, starts with the two benzenoids CgHg (benzene) and 2 12 Then... 

An example of the application of eqn. (13) seems to be warranted, and preferably to a coronoid. However, even the smallest coronoid has an adjacency matrix too voluminous to be reproduced here. Instead we have chosen [lOjannulene, a degenerate coronoid see Chart 1. In this example, eqn. (13) yields... 

It is found by inspection that dicumovalene is nonperforable (by three naphthalene holes). Notice that the heavy-line cycle in the above depiction cannot be trespassed by any corona hole, (iv) = 1, B = Ce7H2i(iTs=24, 31) represents 43 isomers, which were generated and depict probably for the first time. The depictions are not reproduced here, but they were diecked in order to verify that exactly two of these isomers were perforable and led to the two C67H2 (h=18, n l) extremal triple coronoids, which are shown in Fig. 3. (v) h = 19, = 3, A... 

John (1990) has supplied the necessary and sufficient condition for the possibility to circumscribe a benzenoid B an unlimited number of times. When this is possible, B is called a reproducible benzenoid. The mentioned condition was re-formulated in terms of the latent row of hexagons, a possible formation on the perimeter of a benzenoid (Cyvin SJ, Cyvin, Brunvoll, Gutman and John 1993). Below we give the adaptations of this approach to coronoids. 

Observation 5.2 A coronoid C is reproducible if and only if C does not possess any latent row of hexagons. 

The class of reproducible core coronoids (analogous with the reproducible core benzenoids) is especially important. A reproducible core coronoid Cq fulfils the condition that... 

In other words, any extremal single coronoid (A) can be circumscribed without limitation. However, there also exist nonextremal single coronoids which are reproducible and therefore can be circumcscribed without limitation. Kekulene is an obvious example. 

The generating function (76) reproduces the formulas of circular single coronoids (cf. Table 2) as is indicated in (70). The degenerate systems (formulas for s < 16) are excluded. 

Eqn. (7) reproduces the formulas of Table 2, including the two formulas of the degenerate coronoids. 

Proposition 6.1 For every single coronoid formula there exists (at least) one naphthalenic single coronoid C 6 which is reproducible. 

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