Constant-isomer benzenoid series

Constant-Isomer Benzenoid Series General Introduction.88... 

Constant-Isomer Benzenoid Series Preliminary Numerical Results,... 

It was proved that there is no cove and no fjord among the extremal benzenoids. This is a necessary condition for the feature that all such benzenoids can be circumscribed. In general, the absence of coves and fjords is not a sufficient condition for this feature [8]. Nevertheless it is inferred that all extremal benzenoids, say A, can be circumscribed and not only once, but an infinite number of times. We go still a step further. Let (n s) be the formula of A and (ri s ) of drcum-A as in Eq. (11). If all the (n s) benzenoids A are circumscribed, it is inferred that the created circum-A benzenoids represent all the (ri s ) isomers. Then there is obviously the same number of the A and circum-A systems. The properties described in this paragraph are very plausible and represent the foundation of the algorithms [46, 50, 51] and general formulations [51] for constant-isomer benzenoid series, which have been put forward. Also some attempts to provide rigorous proofs are in progress. 

The general expression for the formulas of the ground forms of constant-isomer benzenoid series in the Balaban picture was worked out from the formula (b) of Sect. 8.1. The new pair of parameters <5 was introduced to be consistent with the description in Sect. 8.2. The net result reads [51]... 

In Table 3 a number of formulas for the ground forms of constant-isomer benzenoid series are listed. 

It is useful to have some concrete pictures of the ground forms for constant-isomer benzenoid series, in continuation of Fig. 10 for the one-isomer series. Some of them are displayed in our main reference [8] and more of them elsewhere [20, 43], the latter citation [43] containing detailed documentations to previous works. All the complete sets from these sources are collected in Fig. 11 (without too much overlap with the previous chapter [8]). 

The Kekulean systems belonging to constant-isomer benzenoid series are invariably normal (marked n in Fig. 11) and have A = 0. The non-Kekuleans are marked o with the A value indicated in a subscript (oA). In Fig. 11 also the excised internal structures [8] (see also below) are indicated by contours in bold. The same is found in Fig. 10 and subsequent figures. 

Application of the Special Aufbau Procedure to Ground Forms of Constant-Isomer Benzenoid Series... 

Table 6. Three-parameter codes for (n s) and (n — 3 s — 1), where (n s) represents the ground forms of a constant-isomer benzenoid series...

If C H5 has an excised internal structure [8, 32-34, 38], then its formula is C 2s+6Hs 6 in accord with Eq. (26). This is also the formula which appears in point (c ) of the special aufbau procedure (Sect. 11). In general an excised internal structure may be a non-benzenoid. Several examples of this phenomenon are found in Fig. 10. In connection with Eq. (27), however, when the excised internal structures of not too small ground forms (G) of constant-isomer benzenoid series are going to be considered, we shall only encounter benzenoids as such systems. Therefore, all we need to known about the definition of an excised internal structure here, is if G = circum-G0, then the benzenoid G° is the excised internal structure of G. Let again (n s) be the formula of G, and correspondingly (n° s°) of G°. Then... 

Table 11. Numbers of isomers, classified according to A and symmetry, for constant-isomer benzenoid series with even-carbon atom formulas...

Fig. 17. The smallest ground form of constant-isomer benzenoid series which is an improper snowflake C99oH78 (h = 457)...

Table 13. Numbers of snowflakes among constant-isomer benzenoid series...

Dias [53] has also observed some interesting regularities in the numbers for constant-isomer benzenoid series which combine the even-carbon atom formulas (Table 11) with the odd-carbon atom formulas (Table 12). Here we express these regularities by ... 

Dias JR (1990b) Benzenoid Series Having a Constant Number of Isomers — 2 — Topological Characteristics of Strictly Peri-Condensed Constant-Isomer Benzenoid Series. J Chem Inf Comput Sci 30 251... 

Dias JR (1990e) Constant-Isomer Benzenoid Series and Their Topological Characteristics. Theor Chim Acta 77 143... 

Dias JR (1990g) Topological Characteristics of Strictly Pericondensed Constant-Isomer Benzenoid Series. Z Naturforsch 45a 1335... 

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