Catafusenes

When the title concepts are applied to the simply connected polyhexes we shall sometimes use the brief designations catafusenes and perifusenes [13, 30]. A dualist of a catafusene is acyclic (a tree). 

These numbers, which are defined in Sect. 3.3, pertain to the catacondensed simply connected polyhexes, for short called catafusenes (Sect. 2.6). They include the catacondensed helicenes. 

Unbranched Catacondensed Simply Connected Polyhexes (Unbranched Catafusenes)... 

Explicit formulas for the numbers of the title systems were developed, in a simple combinatorial way, by Balaban and Harary [13] cf. also Balaban [58], The derivation [13] involves a treatment of the subclasses of unbranched catafusenes with specific symmetries. It is outlined in the following, basically in the version of Brunvoll et al. [59] (cf. also Balaban et al. [60]), and is supported by illustrations. 

Let the numbers of unbranched catafusenes belonging to the different symmetry groups be identified by the below symbols. It is stressed that helicenes are included. 

The explicit formulas for s, the number of symmetrical catafusenes, read ... 

All the formulas in the present paragraph also apply to generalizedfibonacenes, viz. unbranched catafusenes consisting of equidistant segments, which are not necessarily 2-segments. 

For the numbers of catafusenes as a function of h, say Ch, an exact asymptotic behavior for large values of h is known [9] and has the form... 

It is recalled that Ch, which pertains to catafusenes, include the catacondensed helicenes Ch are, in other words, the Harary-Read numbers (cf. Sect. 3.3). Based on the form (42), Gutman [67] assumed... 

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. 

Balaban and Harary [9] have a misprint in their number of unsymmetrical (u) unbranched catafusenes with h = 6, while their figure shows correctly all the 16 systems. Another minor misprint in the depiction of unbranched catafusenes with h = 7 by Balaban and Harary [13] one point (representing a hexagon) is omitted. 

The numbers of branched catafusenes (cf. Table 16) are consistent with the differences between appropriate numbers from Table 7 with supplements in text... 

Table 16. Numbers of branched catafusenes benzenoids and helicenes...

Balaban, A.T. and Tomescu, T. (1985). Chemical Graphs. XLI. Numbers of Conjugated Circuits and Kekule Structures for Zigzag Catafusenes and (j, k)-hexes Generalized Fibonacci Numbers. MATCH (Comm.Math.Comp.Chem.), 17, 91-120. 

Balaban, A.T. and Artemi, C. (1989). Chemical Graphs. Part 51. Enumeration of Nonbranched Catafusenes According to the Nmnbers of Benzenoid Rings in the Catafusene and Its Longest Linearly Condensed Portion. Polycyclic Aromatic Compounds, 1,171-189. 

Balaban, A.T. (1993a). Benzenoid Catafusenes Perfect Matchings, Isomerization, Automeriza-tion. Pure ApplChem., 65,1-9. 

A new topological index for catafusenes L-transform of their three-digit codes. Rev. Roum. Chim., 22, 45 7. 

A.T. Balaban, "Enumeration of Catafusenes, Diamondoid Hydrocarbons, and Staggered Alkane C-Rotamers, MATCH (2) (1976), 51-61,... 

Tb orisin Cable iavMvee up to ten bexagoue. The priDtoute Indicate structural formulas of polybexcs, asa>polyhexes, etc. and the totab In the above table ate fuKher subdivided according to the number of Internal vertices, which are 0 for catafusenes and > 0 for perifusenes. 

Figure 9 Bond-rooted catafusenes. (a) 5-catafusene. Only one hexagon adjoins the hexagon with the root bond (thick line), b) D-catafusene. Two hexagons adjoin the hexagon with the root bond.

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