Analytic Properties Of Some Counting Polynomials On Hexagonal Systems

The theoretical research related to the hexagonal system(benzenoid system)mainly originated from the molecular structure of benzenoid hydrocarbons.A hexagonal system is a finite connected plane graph without cut vertices in which every interior face is bounded by a regular hexagon.In the chemical graph theory,the skeleton graph represents only those atoms which can be regarded as forming the framework of the molecule.Other atoms,which are usually hydrogens,are disregarded.The carbon atoms correspond to the vertices of the hexagonal system,and the single or double bonds between carbon and carbon correspond to the edges of the hexagonal system.The carbon-carbon double bonds in benzenoid hydrocarbons correspond to a Kekule structure of the hexagonal system,which is a perfect matching in graph theory.Kekule structure and its related properties are of great significance to the study of the chemical properties of hydrocarbons,such as electronic energy,molecular stability,aromaticity and molecular resonance energy.In this paper,some kinds of counting polynomials on hexagonal systems are considered,including sextet polynomials,Clar covering polynomials,etc.And study their analytic properties by applying analytic tools.The main results can be summarized as follows:1.We consider the analytic properties of forcing polynomials.We show that zeros of the forcing polynomials of pyrene chains are all real.It can also be shown that the coefficients of forcing polynomials of pyrene chains are asymptotically normal.As a result,the coefficients of forcing polynomials of pyrene chains are unimodal and log-concave.2.We consider the analytic properties of Clar covering polynomials.We present some hexagonal systems whose Clar covering polynomials have only real zeros.And use some appropriate methods to determine real zeros of Clar covering polynomials for different situations.We also considered the distribution of real zeros of Clar covering polynomials,and obtained the dense interval of real zeros of Clar covering polynomials.3.We consider the analytic properties of sextet polynomials.For the pyrene chains,we show that zeros of the sextet polynomials of pyrene chains are all real.Therefore,the coefficients of sextet polynomials of pyrene chains are asymptotically normal.As a result,the coefficients of sextet polynomials of pyrene chains are symmetric,unimodal and log-concave.At the same time,the distribution of real zeros of sextet polynomials are also considered.For general hexagonal systems,we show that real zeros of all sextet polynomials are dense in a closed interval.