The Clar Number Of Fragment Geaphene

In 1958,Clar and Zander explained the differences in chemical behaviour of two benzenoid hydrocarbon isomers by the maximum number of mutually resonant hexagons(or disjoint hexagons which all have three double bonds in a Kekule structure)which each of them may contain.This led Clar to define a graphical representation of such a set,later called the Clar formula,and to develop his aromatic sextet theory.Among other results,this theory explains the stability as well as the colour of benzenoid hydrocarbon isomers by their Clar formulas.Many empirical results support Clar's hypotheses and deductions.A graphene fragment is a benzenoid graph which dualist graph is a unicyclic graph.In particular,when the dualist graph of a benzenoid graph is a circle,we call it cyclofusene graph.In this paper,we determine the Clar number of a cyclofusene graph,and prove the bound for the Clar number of graphene fragment.Moreover,we structure the graphene fragment that attain the bound.More precisely,it is shown that the Clar number of a graphene fragment with n hexagons is at most[2n/3].In the first part,the research background of the Clar number is given,the role played by the Clar number in the study of the chemical and physical properties of benzenoid hydrocarbons is described,and some basic concepts needed in this paper are given.Some research results of Clar numbers related to this paper.In the second part,we proved some conclusions about the independent number of subcube unicyclic graphs.In last part,we determine the Clar number of a cyclofusene graph,and prove a bound for the Clar number of the graphene fragment.Moreover,we prove the bound and characterize all extremal graphs with respect to this bound.