Enumeration Of Two Classes Of The Hexacyclic Systems

The theory about hexagonal systems is encouraged, to a certain extent, by the study on the molecular structure of benzenoid hydrocarbon. A hexagonal system, also chemically called a benzenoid system, is a finite and 2-connected subgraph of an infinite planar hexagonal lattice graph. Enumeration related to hexagonal systems is an elementary and interesting problem, which attracts a lot of attention of both mathematicians and theoretical chemists.In this dissertation, we focus our attention on the enumeration of two classes of the so called hexacyclic systems. The whole dissertation consists of three chapters. In the first chapter, we introduce two basic enumeration techniques including Burnside lemma and Polya theorem which will play key roles in our study. In the second chapter, by using Polya theorem and a proper code method, the number of a class of hexacyclic systems is determined. Since this class is closely related to the open-ended nanotubes, the above result could be applied directly to count the number of the open-ended nanotubes. In the third chapter, applying Burnside lemma, we determine the number of a class, named the hollow polygon, of the planar hexacyclic systems with at least one internal vertex.