A Special Form of Numerical Self-Similarity of Kekulé Counts of Benzenoid Hydrocarbons

Previously defined equivalence relation, l, on Kekulé counts, K(B)’s of catacondensed benzenoids, classifies Kekule structures according to the number of their terminal conjugated circuits. The function l, is an enumeration method which uncovers the less transparent combinatorial properties of K(B)’s, such as their numerical self-similarity1, graph generation,2,4 modeling quasicrystals3 and modeling of Feigenbaum’s theory of chaos5. Here, we consider two benzenoid system; Bq, an all-kinked unbranched benzenoid, and B3, an all-kinked benzenoid which has one branched hexagon and for which ali branches are equal which are characterized by the peculiar property that the statistical distribution of the hypercubes (vertices, edges, squares, ...) which constitute their Kekulé spaces remains invariant under the effect of the l function. For these two systems, l is analogous to a percolation process. This property leads to a diagonal equality of conjugated circuit counts of members of the Bo series, which is scaled down by the powers of the golden-mean in the case of the B3 series. K(Bo)’s and/or KCB^j’s are shown to model a one-dimensional quasicrystal