Mathematical properties and structures of sets of sextet patterns of generalized polyhexes

Abstract

Several definitions of sextet patterns and super sextets of (generalized) polyhexes have been given, first by He Wenjie and He Wenchen [1], later by Zhang Fuji and Guo Xiaofeng [2], and by Ohkami [3], respectively. The one-to-one correspondence between Kekulé and sextet patterns has also been proved by the above authors using different methods. However, in a rigorous sense, their definitions of sextet patterns and super sextets are only some procedures for finding sextet patterns and super sextets, not explicit definitions. In this paper, we give for the first time such an explicit definition from properties of generalized polyhexes, and give a new proof of the Ohkami-Hosoya conjecture using the new definition. Furthermore, we investigate mathematical properties and structures of sets of generalized polyhexes, and prove that thes-sextet rotation graphR _s(G) of the set of sextet patterns of a generalized polyhexG is a directed tree with a unique root corresponding to theg-root sextet pattern ofG.