Generation of lattice graphs. An equivalence relation on Kekulé counts of catacondensed benzenoid hydrocarbons

Abstract

An equivalence relation λ is defined on the Kekule space K of branched and/or unbranched catacondensed benzenoid hydrocarbons, K ⊃ {k 1 , k 2, …, k K } where k i is an ith Kekulé structure and K is the number of Kekulé structures of the benzenoid hydrocarbon. The function λ partitions K into subsets whose cardinalities are powers of 2 and can be made to generate regular lattice multicubes. The generation of 0-, 1-, 2-, 3-, 4- and 5-cubes is described. All catacondensed benzenoids which contain up to nine hexagons are investigated and in only seven hydrocarbons does the function λ lead to partition of K in distinct (i.e. unrepeated) powers of 2, and hence coincide with the binary representations of the relevant K values. Furthermore, there is only one isoarithmic pair of benzenoid hydrocarbons for which partition of K is self-conjugate, i.e. symmetric with respect to flipping across the main diagonal. The process of multicube generation is shown to be equivalent to generation of all subsets from a given set. Group-theoretical characters of the individual k i value making a certain lattice (multicube) are demonstrated. This paper describes how the Kekulé structures of catacondensed benzenoid hydrocarbons may be used to generate regular graphs such as the square, the cube, the four-dimensional cube, etc. Further, the results lead to computational and combinatorial implications for K, the number of Kekulé structures of catacondensed benzenoid systems.