Kekule Structures of Toroidal Polyhexes

Sachs and John devised another method in which K is evaluated as a determinant of a size not more than half the number of vertices, and in some special cases simple formulas can be derived for example, to a toroidal polyhex that can be represented as a folded parallelogram with a side having three hexagons, the following result can be applied  

Here q = (number of hexagons)/3 i = 2 if g is odd or 3 ifq is even. Examination of the Ceo toroidal polyhexes shows that one, TPH(30-10-1), conforms to this type. Here q = 30/3 = 10 (even), so that 

The absolute value of the tail coefficient of the characteristic polynomial (which is known to give K for benzenoids ), however, is only 3069 (Table 3), and this apparent discrepancy is another reminder that these 3D structures differ in important ways from simple planar benzenoids. All conjugated eireuits in the latter are stabilizing,but for tori, the closure of the lattice around the tube allows the presenee of destabilizing circuits in some cases. 

The apparent (even if rather deceptive) simplicity of these structures gives the feeling that simple algorithms requiring negligible computation, such as the Gordon-Davison method for unbranched cafa-condensed benzenoids or its reformulated 

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