Analytic Properties Of Some Sextet Polynomials

A hexagonal system(benzenoid system)is a finite connected plane graph without cut vertices,each of whose inner face is a unit regular hexagon.In chemical graph theory,the atomic skeleton diagram of a benzene hydrocarbon corresponds to a hexagonal system after ignoring the hydrogen atoms of the benzene hydrocarbon.The study of sextet polynomials is very important for solving Clar numbers and perfect matching numbers,and it is also convenient to understand some other topological properties.This thesis mainly studies analytic properties of sextet polynomials on hexagonal systems.The thesis is organized as follows:In the first part,we introduce the research background of hexagonal systems and the related basic concepts,and summarize its research progress and ideas.In the second part,we investigate zero problems of sextet polynomials of several types of hexagonal systems.In this part,we use the method of matrix theory to obtain the recursive formulas of the sextet polynomials ?(U_n??)??(V_n??).We show that ?(U_n??)? ?(V_n??),are all real,and investigate the distribution and the dense interval of the zeros of the sextet polynomials ?(F_n??).In the third part,we study the unimodality and log-concavity of the sequence of coefficients of sextet polynomials.In this chapter,the sextet polynomials ?(U_n??)??(V_n??),can be easily proved using the classic Newton inequality.For the sextet poly-nomial ?(F_n??),with complex roots,the recursive relationship is mainly used to obtain a closed form,which further proves the unimodality and log-concavity of its coefficient sequence.