Harary—Harborth picture

Benzenoid (chemical) isomers are, in a strict sense, the benzenoid systems compatible with a formula C H, = (n s). The cardinality of C HS, viz. C HS = n, s is the number of isomers pertaining to the particular formula. The generation of benzenoid isomers (aufbau) is treated and some fundamental principles are formulated in this connection. Several propositions are proved for special classes of benzenoids defined in relation to the place of their formulas in the Dias periodic table (for benzenoid hydrocarbons). Constant-isomer series for benzenoids are treated in particular. They are represented by certain C HS formulas for which n s = In Sjl = n2 52 =. .., where (nk sk) pertains to the k times circumscribed C HS isomers. General formulations for the constant-isomer series are reported in two schemes referred to as the Harary-Harborth picture and the Balaban picture. It is demonstrated how the cardinality n s for a constant-isomer series can be split into two parts, and explicit mathematical formulas are given for one of these parts. Computational results are reported for many benzenoid isomers, especially for the constant-isomer series, both collected from literature and original supplements. Most of the new results account for the classifications according to the symmetry groups of the benzenoids and their A values (color excess). 

In this section, two formulas (n s) for O, the circular benzenoids, are reported, viz. (a) and (b). In (a) an elaborate floor function (17) is present. It is avoided in (b) the floor function therein assumes only the value 1 for s = 0 and vanishes for > 0. But instead, the form (b) contains two parameters (k, e), while (a) contains only one, viz. s. We shall refer to the two formulas for O as being in (a) the Harary-Harborth picture and (b) the Balaban picture, respectively. It is recalled that they originate from the analysis of Harary and Harborth [44] in the case of (a) and from Balaban [48] in the case of (b). Under this scope all the inequalities and deductions from them in the previous chapter [8] are in the Harary-Harborth picture. 

Concluding Remark. According to Cyvin SJ, Cyvin and Brunvoll (1993e) eqns. (44) and (50) may be said to belong to the Harary-Harborth picture and Balaban picture respectively cf. also Cyvin SJ and Brunvoll (1991). These designations are rational since eqn. (44) is based on the analysis of Harary and Harborth (1976), while the derivation of (50) is closely related to Balaban (1971). 

Harary-Harborth Picture. The starting point is eqn. (4.44), where is substituted with the result... 

General formulations for the formulas of the title classes have been worked out. Harary-Harborth Picture. From eqns. (5.62) and (2) it is obtained for the formulas of the... 

The six characteristic shapes of circular benzenoids are recognized as in the case of circular single coronoids (Fig. 4.6), and are again indicated by the parameter e = 0,1, 2, 3, 4, 5. The smallest of the systems, which presently are associated with k = 0, are displayed in Fig. 4 three coronoids and three degenerate coronoids. All the C H formulas under consideration are given (in the Harary-Harborth picture) by... 

The six smallest coronoids of the class under consideration (0) are depicted in Fig. 5. Their formulas are given (in the Harary-Harborth picture) by... 

In the following section we report a substantially more general formulation completely general expressions in the two pictures (of Harary-Harborth and of Balaban) are given, in fact, for the formulas of constant-isomer series, both for the ground forms and the higher members. 

Instead of this elaborate specification of the (tiq 5q) formulas we shall be able to give a full account of them in terms of comprehensive spedfications and explicit expressions. All the three "pictures" are going to be invoked the Harary—Harborth—, the Balaban— and the new picture. But first some additional properties of the systems under consideration are needed. In this connection we shall introduce two subclasses of the ground forms and higher members perfect and imperfect. 

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