Graph generation: quasicrystal-like benzenoid systems

Abstract

An equivalence relation is defined on the terminal rings of alternant and non-alternant hydrocarbons including catacondensed benzenoid systems with zero, one and two branched hexagons, benzocyclobutadienes, benzocy-cloctatetraenes and certain pentalene derivatives. This relation led to the generation of vertex-transitive regular graphs which allow the study of many properties of interest, viz. quasicrystal-like codes, ordering diagrams, build-up (aufbau) processes, combinatorial aspects of the number of Kekule structures and a novel postulate on the stability of these hydrocarbons. The graphs generated represent real chemical species such as cubane and certain transition-metal clusters or they may be thought of as depicting chemical transformations. The work is comparable with the use of permutation matrices as graph generators recently adopted by Randicet al.(M. Randic, W. L. Woodworth, A. F. Kleiner and H. Hosoya, J. Comput. Chem., 1987, 2, 522).