Pericondensed benzenoids

In pericondensed benzenoid systems a Hamiltonian cycle may, but need not exist. The problem of the existence of a Hamiltonian cycle is solved by the following result [11]. 

Theorem 2. A (pericondensed) benzenoid system B possesses a Hamiltonian cycle if and only if there exists an e-transformation, B => B, such that B is catacondensed. 

Eq. (7) does not hold for all pericondensed benzenoids and its range of validity was established in a series of investigations [90, 92-96]. The work of Ohkami [96] can be considered as a complete solution of this problem. 

It is really surprising that the Wiener indices have such number-theoretical properties. This kind of modular behavior was observed never before for any of the numerous topological indices studied in chemical graph theory. Results analogous to those given in Theorem 20 were later found for other classes of (non-benzenoid) graphs [130, 131], but their extension to pericondensed benzenoids was never accomplished. 

Numbers for benzenoids separately can be determined as the differences between appropriate numbers in Tables 1 and 2, but this is not the way they were deduced originally. The Dusseldorf-Zagreb numbers (or DZG numbers) is the designation which has been used [16,17] about the data for eatacondensed and pericondensed benzenoids, and benzenoids in total with h < 10. These data have later been supplemented by other investigators [19,46, 47], as well as the Dusseldorf-Zagreb... 

The Dusseldorf-Zagreb numbers, in the original sense, are the numbers of catacondensed and pericondensed benzenoids for h < 10 and their sums, which give the numbers of benzenoids in total. The actual numerical values (for the catacondensed systems and totals) are found in different places [15,17,53, 76], and in particular they are reproduced in the book of Knop et al. (Dusseldorf-Zagreb group) [16]. The numbers are displayed in Table 23. 

For the numbers of benzenoids with very small h values the documentation is again very difficult and ambiguous cf. Sect. 3.5. In Table 23 we have followed the consolidated report [18] and supplemented it with a quotation for the pericondensed benzenoids (cf. footnotes to the table). In addition, we have chosen to give credit to Harary [2] for the separate numbers of catacondensed and pericondensed benzenoids with h < 4 because he, at an early date, depicted all the forms of these systems. 

In the following we point out specifically some errors, apparently all of them misprints. They may seem to be a trifle, but should nevertheless be treated seriously. In the paper of Aboav and Gutman [69] the number of C 13 (from Cioslowski) is claimed to be wrong [47]. The same (wrong) number was quoted in a table of the monograph by Gutman and Cyvin [22], but in parentheses as uncertain. The parenthesized number for pericondensed benzenoids with h = 13 is also wrong. Furthermore, in the same table, there is a misprint in C g and in the total for h = 15, the last number therein. 

As far back as 1968, Balaban and Harary [13] were aware of the unique position of zethrene, which was placed in a class of its own under the pericondensed benzenoids with h = 6. It is an essentially disconnected benzenoid. However, these authors did not sort out the other two essentially disconnected benzenoids (annelated perylenes) with the same numbers of hexagons. Neither did they sort out perylene itself, which is the unique essentially disconnected benzenoid with h = 5. In the table we are referring to [13], the entry for the classified pericondensed system with h = 3 is misplaced. 

The numbers of catacondensed benzenoids belonging to the symmetries Dlh and C2h are listed in Tables 14, 15, 17 and 20. When taking the catacondensed and pericondensed benzenoids of these symmetries together, most of the relevant information on the numbers of dihedral (Dlk) and eentrosymmetrical (C2h) benzenoids, which is available so far, is already contained above cf. Tables 26, 27, 28 and 30. Collections of the specific data for the D2h and C2h systems are... 

Tables 41 and 42 give an account of the classification according to symmetry for the catacondensed and pericondensed benzenoids, respectively. Most of the data are from Cyvin etal. [127] cf. also Gutman and Cyvin [22], For the catacondensed all-benzenoids with D2h symmetry, we have also generated by hand, in continuation of Table 41, 3 systems with h = 31, and 13 systems with h = 43, claiming that this covers all such systems for h < 43.

Strictly Pericondensed Benzenoid and Excised Internal Structure. . . 200... 

The pericondensed extreme-left benzenoids constitute a subclass of the strictly pericondensed benzenoids in the sense of Dias [12, 15, 19, 21, 55-57] they are defined by having all their internal vertices connected and no catacondensed appendages. Phenalene, C13H9, which has only one internal vertex, is reckoned among the strictly pericondensed benzenoids. An equivalent definition in a most succint form reads ... 

A strictly pericondensed benzenoid is a benzenoid with h > 2 and all its internal... 

The formulas at the extreme left in the periodic table for benzenoid hydrocarbons, except C10H8 (naphthalene), represent exclusively strictly pericondensed benzenoids [12, 16, 21], viz. the pericondensed extreme-left benzenoids. Formulas for ds< 0 and not at the extreme left represent both strictly pericondensed and non-strictly pericondensed benzenoids. For ds > 0 there are no strictly pericondensed benzenoids [56]. 

It should be clear that strictly pericondensed benzenoids occur for formulas at unlimited distances from the staircase-like boundary. Consider, for instance, the homolog series of hydrocarbons as shown in Fig. 2. The systems have in general (for h 2, s > 8) the formulas ... 

The extremal benzenoids have no coves and no fjords, but this property is not valid for strictly pericondensed benzenoids in general. Figure 3 shows some counterexamples. 

The excised internal structure [10,12,19,21,22,55] is defined in connection with strictly pericondensed benzenoids. It is the set of internal vertices and the edges... 

Fig. 3. The smallest strictly pericondensed benzenoid with a cove, C31H15, which is non-Kekulean (A = 1). The two smallest Kekulean strictly pericondensed benzenoids with a cove, C34H16, which are normal. The smallest strictly pericondensed benzenoid with a fjord, C40H18, Which is normal (Kekulean)...

The numbers C4/,+2H2 +4 for h > 1 increase steadily with the h values, for 5 < /i < 15 by a factor between three and four. On the other hand, if the pericondensed benzenoid hydrocarbons (corresponding to benzenoids with internal vertices) are taken into account, the numbers of isomers may seem to vary in a chaotic way. For all the isomers with H14 in the formula, for instance, one has the set of numbers ... 

Several of these properties are illustrated in Fig. 6. In many cases the scheme (a) alone is sufficient and is then indicated by a vertical arrow alone. That applies to most of the formulas for pericondensed benzenoids in Fig. 6. For C24H14, C28H14, C31H15, C34H16, C36Hi6 of Fig. 6 and C4iH17 of Fig. 7 it is indicated, by one horizontal and one vertical arrow pointing to each box, that the schemes... 

Propositions 1 (a) The staircase-like boundary has no one-formula step. (b) The staircase-like boundary has no plateau of more than one formula for pericondensed benzenoid (s). 

The above relation hcdds for some, but not for all pericondensed benzenoids. Necessary and sufficient conditions which a pericondensed benzenoid system must obey in order that 5ez(G,l) = K have been recently established by Zhang and Chen [179]. 

Dias JR (1991d) Strictly Pericondensed Benzenoid Isomers. Match 26 87... 

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