Reciprocal polycycle

Consider now the notion of reciprocity, defined for some proper polycycles. Let P be a proper bounded (r, q)-polycycle. Consider the union of all r-gonal faces of r, q outside of P. Easy to see that this union will be an (r, 4)-polycycle call it then a reciprocal polycycle to P if either P is elliptic or P is infinite and has a connected boundary. Call a polycycle self-reciprocal if it admits the reciprocal polycycle and is isomorphic to it. 

The term diversity-oriented synthesis (DOS) is relatively new and, as mentioned above, is usually defined as the synthesis of complex, natural product-like molecules using a combinatorial approach and employing the full palette of modern organic reactions. It may be a subject of discussion what exactly qualifies a molecule as being natural product-like [4], and in most cases the similarity to an actual natural product seems reciprocal to the number of synthesized compounds. However, even in less complex cases, the products may be highly substituted polycyclic structures with defined stereochemistry, reminiscent of natural products [19, 20]. In these cases, a moderately complex backbone structure is subsequently modified with a well-established set of selective reactions to introduce diversity. 

For (5,3)-polycycles with x < 11, Ns x) was found in [CCBBZGT93] all such extremal (5,3)-polycycles turn out to be proper and unique. Moreover, the (S, 3)-polycycles, which are reciprocal (see Section 4.1) to any such extremal ones, turn out to be also extremal. [CCBBZGT93] asked about N ix) for x > 12 this section answers this question for any x and for all elliptic (p, q). 

Bonchev, D., Balaban, A.T., Liu, X. and Klein, D.J. (1994a). Molecular Cyclicity and Centricity of Polycyclic Graphs. I. Cyclicity Based on Resistance Distances or Reciprocal Distances. Int.J. Quant. Chem., 50,1-20. 

This equation, as a double reciprocal plot, is similar to Eq. (5) but is applied to probe mobility, not quencher dynamics. The same values for the dissociation rate constants (Table 17) were obtained when employing these two different methods with quenchers in different phases, suggesting that the underlying assumptions for the derivation of Eqs. (27) and (28) were reasonable. The entry rate constants were diffusion controlled, and the exit rate constants varied by a factor of 3. In analogous fashion to the polycyclic aromatic hydrocarbon probes, the exit rate constants were faster for the more polar ketones p-methoxyacetophenone and acetophenone compared to isobutyro-phenone or propiophenone [193,194],... 

D. Bonchev, A.T. Balaban, X. Liu, and D.J. Klein, Molecular cyclicity and centricity of polycyclic graphs. I. Cyclicity based on resistance distances or reciprocal distances, Int. J. Quantum Chem. 50 (1994) 1-20. 

Quinine (QN) and quinidine (QD) are natural enantiomers belonging to the cinchona alkaloid family, an important subgroup of naturally occurring polycyclic P-carboline alkaloids. They are widely used as resolving agents for chiral acids via preferentially forming diastereomerie salts with one of the enantiomers [117]. From the reciprocity concept point of view, the logic path is that QN and QD are potential chiral selectors... 

---