Polyhex

HarF70 Harary, F., Read, R. C. Enumeration of tree-like polyhexes. Proc. Edin. Math. Soc. Ser. II 17 (1970) 1-13. 

Cyvin BN, Brunvoll J, Cyvin SJ (1992) Enumeration of Benzenoid Systems and Other Polyhexes. 162 65-180... 

Spherical and toroidal fullerenes have an extensive chemical literature, and Klein bottle polyhexes have been considered, for example, in [Kir97, KlZh97]. 

Figure 3.1 Smallest spherical, toroidal, Klein bottle and projective fullerenes. The first column lists the graphs drawn m the plane, the second the map on the appropriate surface and the third the dual on the same surface. The examples are (a) Dodecahedron (dual Icosahedron), (b) the Heawood graph (dual Ky), (c) a smallest Klein bottle polyhex (dual 3,3,3), and (d) the Petersen graph (dual Ke).

At least one toroidal polyhex that is cell-complex exists for all numbers of vertices v > 14. The unique cell-complex toroidal fullerene at v = 14 is a realization of the Heawood graph. It is GC2,i(hexagon) in terms of Goldberg-Coxeter construction and is the dual of K7, which itself realizes the 7-color map on the torus. This map and its dual are shown in Figure 3.1. 

There is a literature (see, for example, [GrSh87a, Section 9.4], [BGOR99], [BCH02], and [BCH03]) about proper parabolic polycycles (polyhexes, polyamends, polyominoes for 6,3, 3,6), 4,4, respectively) the terms come from familiar terms hexagon, diamond, domino, where the last two correspond to the case p, ... 

Polyhexes are used widely (see, for example, [Dia88, Bal95]) in organic chemistry they represent completely condensed PAH (polycyclic aromatic hydrocarbons) C Hm with n vertices (atoms of the carbon C), including m vertices of degree two, where atoms of the hydrogen H are adjoined (see Figure 7.1). 

BCH03] G. Brinkmann, G. Caporossi, and P. Hansen, A survey and new results on computer enumeration of polyhex and fusene hydrocarbons, Journal of Chemical Information and Computer Science 43-3 (2003) 842-851. 

BCC92] J. Brunvoll, B.N. Cyvin, and S. J. Cyvin, Enumeration of benzenoid systems and other polyhexes, Topics in Curr. Chem. 162 (1992) 65-180. 

HLZ96] P. Hansen, C. Lebatteux, and M. Zheng, The boundary-edges code for polyhexes, Journal of Molecular Structure (Theochem) 363 (1996) 237—247. 

Since computer enumeration methods are still hampered by the explosive growth in the number of polyhex isomers having more than eleven hexagonal rings, the most practical approach is the enumeration of select benzenoid groups of interest to chemists [8],... 

If two polyhexes have the same number of edges q, then they must be isomers. Note that the number of edges is odd for every other column in the Formula Periodic Table for Benzenoid Hydrocarbons (PAH6s). Thus benzenoid hydrocarbons having formulas belonging to the ds —. .., — 2,0,2,4,... column series have q = odd number. 

For a given polyhex graph one can draw a set of sextet patterns with various numbers of resonant sextets including the zero-sextet pattern. Let the number of the sextet patterns of G with k resonant sextets be denoted as r(G, k). The total number of the sextet patterns s2j is... 

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