GENERATION OF LATTICE GRAPHS - AN EQUIVALENCE RELATION ON KEKULE COUNTS OF CATACONDENSED BENZENOID HYDROCARBONS

An equivalence relation lambda is defined on the Kekule space K of bra nched and/or unbranched catacondensed benzenoid hydrocarbons, K supers et-of {k1, k2,..., k(K)} where k(i) is an ith Kekule structure and K i s the number of Kekule structures of the benzenoid hydrocarbon. The fu nction lambda partitions K into subsets whose cardinalities are powers of 2 and can be made to generate regular lattice multicubes. The gene ration of 0-, 1-, 2-, 3-, 4- and 5-cubes is described. All catacondens ed benzenoids which contain up to nine hexagons are investigated and i n only seven hydrocarbons does the function lambda lead to partition o f K in distinct (i.e. unrepeated) powers of 2, and hence coincide with the binary representations of the relevant K values. Furthermore, the re is only one isoarithmic pair of benzenoid hydrocarbons for which pa rtition of K is self-conjugate, i.e. symmetric with respect to flippin g across the main diagonal. The process of multicube generation is sho wn to be equivalent to generation of all subsets from a given set. Gro up-theoretical characters of the individual k(i) value making a certai n lattice (multicube) are demonstrated. This paper describes how the K ekule structures of catacondensed benzenoid hydrocarbons may be used t o generate regular graphs such as the square, the cube, the four-dimen sional cube, etc. Further, the results lead to computational and combi natorial implications for K, the number of Kekule structures of cataco ndensed benzenoid systems.