Catacondensed benzenoid

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). 

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. 

This type of Clar structure can be generated for a limited number of classes of catacondensed benzenoid hydrocarbons by defining an operation which resembles coloring [14] of the vertices of certain caterpillars but allows more flexibility. Namely, we define a Clar coloring [ 15] of the vertices of a caterpillar by the following steps ... 

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

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. 

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. 

A further method for calculating the sextet polynomial of an unbranched catacondensed benzenoid molecule, not based on the Gutman-tree concept, was reported in [113]. 

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... 

Section 6 deals with the enumeration of catacondensed simply connected polyhexes, i.e. catacondensed benzenoids and helicenes. 

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. 

A Kekulean benzenoid may be normal or essentially disconnected. In an essentially disconnected benzenoid there are fixed double and/or single bonds. A fixed single (resp. double) bond refers to an edge which is associated with a single (resp. double) bond in the same position of all the Kekule structures. A normal (Kekulean) benzenoid has no fixed bond. All catacondensed benzenoids are normal therefore all essentially disconnected benzenoids are pericondensed. But a pericondensed Kekulean may be either normal or essentially disconnected. 

Fig. 12. Unbranched catacondensed benzenoids representing all Gutman trees (or LA-sequences) for h < 8...

For the number of unbranched catacondensed benzenoids, U h, Gutman [61] launched the very simple approximate formula... 

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. 

Let the numbers of unbranched catacondensed benzenoids and unbranched catacondensed helicenes be denoted by V and [/, respectively. Then... 

Table 14. Numbers of unbranched catacondensed benzenoids unbranched catacondensed helicenes in parentheses +...

Table 15. Numbers of symmetrical unbranched catacondensed benzenoids corresponding helicenes in parentheses+...

The unbranched catacondensed benzenoids (without helicenes) up to h = 7 are represented as dualists in Fig. 14. These forms were first given by Balaban and... 

Table 17. Numbers of branched catacondensed benzenoids, classified according to symmetry...

In Table 17, a detailed account on the numbers of branched catacondensed benzenoids is displayed, including the distribution into symmetry groups. Here the numbers for D3h and C3h at h < 13 are also obtainable from a scrutiny of figures published by Cyvin et al. [78]. Supplements to Table 17 are found in or from some of the subsequent tables. 

In Fig. 16 all the branched catacondensed benzenoids (without helicenes) up to h = 7 are given in the dualist representation. They have been given for h < 6, with helicenes included, by Balaban and Harary [13] and by Balaban [73], The forms for h = 7 are found in an other paper by Balaban [30], The above mentioned works of Dzonova-Jerman-Blazic and Trinajstic [74], by Gutman [75], Trinajstic et al. [76] and by El-Basil [77] display the figures of both branched and unbranched benzenoids, with h < 6, h < 5, h = 1 and h < 5, respectively. 

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... 

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