The Restricted-Subduced-Cycle-Index (RSCI) Method for Symmetry-Itemized Enumeration of Kekulé Structures and Its Application to Fullerene C60

Abstract

Abstract The restricted-subduced-cycle-index (RSCI) method for symmetry-itemized enumeration of Kekulé structures has been developed as an extension of the unit-subduced-cycle-index (USCI) approach (S. Fujita, “Symmetry and Combinatorial Enumeration in Chemistry,” Springer-Verlag, 1991). In the RSCI method, such Kekulé structures are considered to be generated by edge substitution of a skeleton under a restricted condition that no selected edges have a common vertex. The restricted condition is formulated by means of the concept of territory indicators and territory discriminants, by which subduced cycle indices (SCIs) derived from unit subduced cycle indices (USCIs) are converted into restricted subduced cycle indices (RSCIs). The resulting RSCIs are used to evaluate marks (numbers of fixed points) in place of the SCIs used in the USCI approach, so that the numbers of Kekulé structures are calculated in a symmetry-itemized fashion. Moreover, the resulting data of enumeration are used to draw respective Kekulé structures by means of orbit indicators and edge indicators. The application of the RSCI method to fullerene C60 of Ih-symmetry is discussed in detail.