Some Structural Characters Of Essentially Disconnected Coronoid Systems

Hexagonal systems (i.e. benzenoid system) and coronoid systems are the natural representations of the skeletons of molecules of benzenoid hydrocarbons and coronoid hydrocarbons in the field of organic chemistry. A hexagonal system H is a finite connected subgraph of the infinite hexagonal lattice without cut vertices or nonhexagonal interior faces. A coronoid system G is the subgraph of a benzenoid system H which has at least a nonhexagonal interior face, and each edge belongs to some hexagon of G. A Kekuléan polyhex is either a benzenoid system with Kekulé structure or a coronoid system with Kekulé structure. The term "essentially disconnected" was used for the first time by Cyvin et al. to indicate those Kekuléan polyhexes with fixed bonds(single or double). Kekuléan polyhexes without fixed bonds are referred to as normal. An essentially disconnected coronoid system is naturally defined as a coronoid system which has fixed bonds and has at least one nonhexagonal interior face. The present paper mainly studies an important project in chemistry graphy: the structural character of essentially disconnected coronoid systems whose research and description has an very important role in the research of coronoid hydrocarbons. An essentially disconnected benzenoid system is defined as a benzenoid system which has fixed bonds. For an essentially disconnected benzenoid system, it has detailly been proved that such a subgraph obtained by deleting all the fixed single bonds and all the end vertices of the fixed double bonds, has at least two components, and that each component is also a normal benzenoid system. Therefore, we further considerate essentially disconnected coronoid systems and essentially disconnected Kekuléan polyhexes. Finally, we do some researches to the feature of boundary vertices of coronoid systems. The main results of the paper are as followings:An essentially disconnected coronoid system G, after deleting all the fixed single bonds and all the end vertices of the fixed double bonds of G, is disconnected , and has at least two normal components.An essentially disconnected Kekuléan polyhex H, after deleting all the fixed single bonds and all the end vertices of the fixed double bonds of H, is disconnected , and has at least two normal components.Let G be a coronoid system, be the number of vertices with degree 2 on the boundary of G, be the number of vertices with degree 3 on the boundary of G, then , where k is the number of nonhexagonal interior faces of G; especially for a single coronoid system G, then . Meantime, let H be a generalized single coronoid system H, then .