Catacondensed benzenoid system

A catacondensed benzenoid system (CBS) is a BS in which no vertex belongs to more than two cells. Two cells of a BS are adjacent if they have an edge in common. The dualist graph D = D(B) of a benzenoid system B has as its vertices the centers of the cells of B where two vertices of D are connected by a straight line segment if and only if the corresponding cells are adjacent in B. 

If the number of vertices is even, then an evidently sufficient condition for the existence of Kekule structures of a benzenoid system is the existence of a Hamiltonian path [17]. As a corollary, all catacondensed benzenoid systems are Kekulean [18]. But the condition is not necessary. A Kekulean benzenoid system with no Hamilton path is shown in Fig. 4. 

Since all catacondensed benzenoid system are normal, we may reduce a generalized benzenoid system with fused catacondensed units without altering the Kekulean/non-Kekulean character. By inspecting Fig. 19. the reader may get a fair idea about how the reduction is made. 

Corollary 1.1. Catacondensed benzenoid systems do not possess cycles whose sizes are divisible by four. 

Since a catacondensed system possesses no internal vertex its perimeter embraces all the vertices. Consequently, the perimeter of a catacondensed system is a Hamiltonian cycle. In other words, all catacondensed benzenoid systems are Hamiltonian. 

The first exact result obtained along these lines was Theorem 10 [91]. Eq. (7) holds for all catacondensed benzenoid systems. 

Corollary 15.3. If B is an unbranched catacondensed benzenoid system then all the zeros of its sextet polynomial are real and negative numbers. 

The above conjecture is true if B is a catacondensed benzenoid system and Z is its perimeter. Clearly, the size of Z is then equal to n = 4h + 2 whereas... 

Table 18. Numbers of different special catacondensed (benzenoid) systems (SCS s) ...

Fig. 4. The code of Kekule structure of catacondensed benzenoid system containing 4 terminal hexagons. Compare also Fig. 3...

Sheng RQ (1991) The Number of Kekule Structures of Unbranched Catacondensed Benzenoid Systems and Primitive Coronoid Systems. Match 26 247... 

Algorithms B and D enable the same to be done, in a more efficient way, for those benzenoid systems whose dualist graph is a tree (representing catacondensed benzenoid hydrocarbons). 

In other words, a cycle in a catacondensed benzenoid is of the size 6 or 10 or 14 or 18. .. This is because catacondensed systems (by definition [3]) possess no internal vertices. On the other hand, according to Theorem 1 the existence of a cycle whose size is divisible by four implies the existence of at least one internal vertex. 

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. 

In addition to the long-known results for M0, M2 and M4 (i.e. b(B, 0), b(B, 1) and b(B, 2)) [2], Dias [39] and Hall [42] discovered the actual form of the dependence of M6 on the structure of a benzenoid system. Dias [39] also reported a formula for b(B, 4), valid for catacondensed systems only. Formulas for Mg and M10 of arbitrary benzenoids and for M12 of catacondensed systems were obtained quite recently [45]. The first few of these expressions read ... 

Corollary 14.1. A benzenoid system B is normal if, and only if, (a) B is catacondensed or (b) the excised internal structure of B possesses a perfect matching. 

In addition to these, only a limited number of other topological indices of benzenoid molecules have been studied. With a few not too important exceptions, generally valid mathematical results were obtained only for one of them — namely for the Wiener index. Therefore the remaining part of this section is devoted to the Wiener index of benzenoid systems. (Further graph invariants worth mentioning in connection with benzenoids, especially unbranched catacondensed systems, are the Hosoya index [119-121], the Merrifield — Simmons index [122, 123], the modified Hosoya index [38] and the polynomials associated with them.)... 

Recurrence relations have been established for the Wiener index of cataconden-sed benzenoid systems, both unbranched [125] and branched [126]. Based on these relations it could be shown that for unbranched catacondensed systems Uh with h hexagons [127],... 

The planar (non-helicenic) circulenes have also been considered separately, first by the Diisseldorf-Zagreb group [16,17, 35] see Table 5. He and He [33] depicted the forms of these systems for h <, 9 from these figures we have extracted separate numbers for the catacondensed and pericondensed systems up to this h value. For the catacondensed planar monocirculenes with h < 11 the numbers may be deduced from a work of Brunvoll et al. [54], who enumerated the benzenoid systems composed of coronene with catacondensed appendages up to h = 12. On... 

Another subdivision of benzenoids (apart from catacondensed/pericondensed) distinguishes between Kekulean and non-Kekulean systems. A Kekulean benzenoid system possesses Kekule structures (K > 0). A non-Kekulean benzenoid has no Kekule structure (K = 0). The shorter designations Kekuleans and non-Kekuleans are often used. All catacondensed benzenoids are Kekulean therefore all non-Kekuleans are pericondensed. Any Kekulean benzenoid has a vanishing color excess A = 0. 

In the present section the catafusenes (catacondensed simply connected poly hexes cf. Sect. 5.1) are treated. However, in contrast to the Harary-Read numbers (first column of Table 7) we shall be interested in the numbers of unbranched and branched systems separately. The numbers of unbranched catafusenes (Table 10) are known from algebraic formulas (cf. Sect. 5.2), but now we are interested in the unbranched catacondensed benzenoids and helicenes separately. Likewise we shall treat the numbers of branched catacondensed benzenoids and helicenes separately. 

After the definition and enumeration of different special catacondensed systems (SCS s) the catacondensed benzenoids belonging to the symmetries D3h, C3h, D2h and C2h are treated in particular. Those of the D3h and DZh symmetries were enumerated by an algorithm invoking SCS s. 

Finally some results for unbranched catacondensed benzenoids with equidistant segments are reported. These systems are the benzenoids (without helicenes) belonging to fibonacenes and generalized fibonacenes. 

Tosic et al. [79] defined the title systems in the course of a particularly efficient algorithm for enumerations of branched catacondensed benzenoids with regular trigonal (D3h) symmetry. The SCS s are (unbranched and branched) catacondensed benzenoids defined in such a way that isomorphic systems of this kind may be reckoned as different , depending on their orientation with respect to an axis. The counting of different SCS s is therefore not a single counting of non-isomorphic... 

In Fig. 18 the forms of the Dik catacondensed benzenoids up to h = 25 are displayed. They have been given previously [79], Being catacondensed, all these systems are normal and therefore Kekulean. The Kekule structure counts (K) are given in the figure. 

A catacondensed benzenoid with dihedral symmetry, viz. D2h is either a branched system or an (unbranched) linear acene. A centrosymmetrical (C2h) catacondensed benzenoid is either branched or unbranched. The D2h systems under consideration have been enumerated by the efficient algorithm invoking SCS s (cf. Sect. 6.4) [80], Table 20, in combination with Table 17, shows the known numbers for the branched catacondensed Dlh and C2h benzenoids. The numbers of unbranched catacondensed benzenoids with C2h symmetry are found under the designation d in Tables 14 and 15 for h < 20 and 21 < h < 30, respectively. 

Figures 20 and 21 display the forms of the unbranched catacondensed benzenoids with only 2-sements or 3-segments, respectively, up to the systems with 7 segments.

For benzenoids with k = 12 the first data were published in 1988 as a result of a collaboration between He He and Trondheim (cf. Sect. 2.8 for a listing of the research centres). The number C l2 (for catacondensed benzenoids with h = 12) became available after the enumeration of the branched catacondensed h = 12 systems in Trondheim, while He He succeeded in a complete enumeration of benzenoids with this number of hexagons. Consistent h = 12 numbers were reported [69] as private communications from He He and from Cioslowski. 

The numbers of catacondensed benzenoids of trigonal symmetry are listed in Table 17 with a continuation in Table 19. A listing for all benzenoids of trigonal symmetry (catacondensed + pericondensed) is given in Table 34, which supplements Table 26. Tables 35 and 36 take into account various divisions into subclasses for the D3h and C3h systems, respectively. The total numbers of D3/t(ia) and of Z>3fc(ib) systems have been given elsewhere [110] and are consistent with those of Table 35. 

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... 

In the first enumerations of catacondensed polyhexes [32, 34, 35] the helicenic systems are included. The smallest helicenic system, viz. hexahelicene, occurs for h — 6. However, from the forms of generated catacondensed benzenoids depicted in some of the early works cited above [32, 34] and others [23, 36-41] the numbers of catacondensed benzenoids (without helicenes) for h up to 7 (C7) are easily extracted. Specific documentations are found in Tables 1 and 2. 

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