Theoretical Chemistry Accounts

, Volume 113, Issue 2, pp 73–79 | Cite as

Graphs to chemical structures 1. Sphericity indices of cycles for stereochemical extension of Pólya’s theorem

  • Shinsaku FujitaEmail author


Pólya’s theorem has been concluded to be concerned with graphs, but not with chemical structures, where it is incapable of treating chiral ligands properly. In order to take account of chiral ligands along with achiral ones, coset representations (CRs) for cyclic subgroups have been examined to classify permutations of the CRs into proper and improper elements. As a result, a k-cycle contained in each permutation has been classified into an enantiospheric, homospheric, or hemispheric one. Thereby, sphericity indices of k-cycles have been defined according to the enantiospheric, homospheric, or hemispheric nature of each k-cycle. On the basis of the sphericity indices, cycle indices with chirality fittingness (CIs-CF) have been defined in place of Pólya’s cycle indices. The CIs-CF have been proved to be capable of enumerating of stereoisomers with chiral and achiral ligands. Their capabilities have been confirmed by using allene derivatives as examples.


Cycle index Chirality fittingness Sphericity index Pólya’s theorem 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of Chemistry and Materials TechnologyKyoto Institute of TechnologyKyotoJapan

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