The Relationship Between The Count Of AKS And GKS Of Coronoid System

A hexagonal system is a connected plane graph without cut vertices in which each interior face is a regular hexagon of side length one.A geometric Kekule Structure(GKS)of a hexagonal system is equivalent to a perfect matching,which corresponds to an algebraic Kekule Structure(AKS).In fact,it is a function that assigns to each hexagon of a hexagonal system one integer:each double bond in GKS that belongs to only one hexagon contributes 2 to that hexagon and each double bond that is shared by two hexagons contributes 1 to each one of these two hexagons.So the function value on each hexagon is the sum of the contribution of the double bond in it.Let us call it Randic number or ?-electron number.Gutman et al.proved that there exists an one-to-one correspondence between GKS and AKS of any cata-condensed hexagon system(no three hexagon share a common vertex)with at least 2 hexagons.But the result does not always hold.T.Balaban et al.also considered the distribution of ?-electrons on hexagons of coronoid systems.A coronoid system is a connected subgraph G of a hexagonal so that each edge belongs to a hexagon of G and G contains at least one non-hexagonal inner face(we called a hole).We call it cata-condensed coronoid if no three hexagon share a common vertex.A cata-condensed coronoid is unbranched(or primitive coronoid),if each hexagon is adjacent to exactly two hexagons.Otherwise it is branched.This paper mainly considers the relationship between the count of the GKS and AKS of a single hole cata-condensed coronoid.We demonstrated that if the primitive coronoid is connected by L2 and A2 mode alternately or connected by all A2 modes,then the number of AKS is equal to the number of GKS minus 2.The number of AKS is equal to the number of GKS minus 1 for other primitive coronoid.We obtained the following results for branched cata-condensed coronoid:(1)the number of AKS is equal to the number of GKS minus the product of each perfect matching of all branches if it connected by L2 mode,A2 mode alternately and added some branches;(2)the branched cata-condensed coronoid that con-nected by all A2 modes and added some branches,the results is same as(1)if branches belong to same class,there exists an one-to-one correspondence between AKS and GKS if branches belong to different class;(3)there exists an one-to-one correspondence between AKS and GKS for other branched cata-condensed coronoid.