The Unimodality Of The Clar Covering Polynomials Of The Hexagonal Chains And Cyclic Hexagonal Chains

P(H, ω), the Clar covering polynomial of the hexagonal system H, is firstly proposed by Heping Zhang and Fuji Zhang in1993which can unify some impor-tant topological indices of the hexagonal systems, such as the number of Kekule structures σ(H,0), first Herdon number σ(H,1), Clar number C(H), etc. They also posed the following conjecture in1996:The coefficients of P(H, ω) are uni-modal. They showed that the conjecture is true for any hexagonal systems H with1≤C(H)≤5at the same time. In this paper, we show the conjectrue is true for hexagonal chains by the relationship between the Clar covering polynomial of a hexagonal chain and matching polynomial of its Gutman tree and the means of log-concave. Then by generalizing this method, we obtain that the Clar cov-ering polynomial of any cyclic hexagonal chain with even segments which is the spreading of the primitive coronoid is unimodal, thus we get some relevant results about primitive coronoids. At last we prove that the Clar covering polynomial of a zigzag cyclic hexagonal chain with odd segments is unimodal by its explicit expression.