Hexagonal Systems With The One To One Correspondence Between Geometric And Algebraic Kekule Structures

For any geometric Kekule Structure(GKS) of a hexagonal system H, we can define an algebraic Kekule Structure (AKS) according to this geometric Kekule Structure. In fact, it is a function that assigns to each hexagon of H one integer according to the following way: each double bond in GKS that belongs to only one hexagon contributes2to that hexagon and each double bond that is shared by two hexagons contributes1to each one of these two hexagons. The concept of algebraic Kekule Structure was introduced by Randic, then many papers studied algebraic Kekule Structure and obtained its many useful properties. When AKS is one-to-one correspondent to GKS, AKS is much more useful.The whole thesis can be divided into four chapters. In the first chapter, we in-troduce AKS’s research background, research meaning, and current research devel-opment. Then we introduce some concepts, terms and marks of this thesis. Finally, we summarize the main results of the thesis.In the second chapter, we study the correspondence between AKS and GKS of Benzenoid parallelogram Bp,q. Vukicevic obtained a necessary and sufficient con-dition that a hexagonal system doesn’t have one-to-one correspondence between its algebraic and geometric Kekule Structures. But this condition can’t provide us with some substantial help, there is still a need for us to study the one-to-one correspon-dence problem between AKS and GKS of hexagonal systems further. In this thesis, we study the necessary and sufficient condition that provided by Vukicevic. and ob-tain a useful lemma, based on this lemma, we obtain that there exists an one-to-one correspondence between GKS and AKS of Benzenoid parallelogram Bp,q, except B2,2and B1,1.In the third chapter, we obtain a property of the pairwise disjoint circles in the theorem obtained by Vukicevic, then based on these properties, we obtain that there exists an one-to-one correspondence between AKS and GKS of any hexagonal system H with no B2,2as its subgraph, except B1,1.In the fourth chapter, we obtain some results about hexagon hexagonal system, and present a conjecture.