Log-concavity Of Independence Polynomials Of Some Kinds Of Trees

The independence polynomials of graphs belong to an important part of algebraic graph theory.In addition,the unimodality properties of independence polynomials are one of the hot topics in algebraic graph theory.In 1987,Erd ?os et al.conjectured that the independence polynomial of any tree or forest is unimodal.Though it attracts many experts' research interests and has some progress,the conjecture is still open.In this thesis,by giving the factorization of independence polynomials and factor the independence polynomials into the product of some factors,we prove the log-concavity of some composite graphs.This provides more examples to support the conjecture of Erd ?os et al..The main results can be summarized as follows:In Chapter 1,introduce the basic concepts of graph theory and some known results of independence polynomials.In Chapter 2,we define new composite graphs called generalized centipede graphs.By solving the recurrence relations,we give the explicit expressions of the independence polynomials.In addition,we consider three special kinds of the generalized centipede graphs belonging to trees and show that their independence polynomials are log-concave and therefore unimodal.In Chapter 3,we define a kind of trees called double-light trees.Using the clawfree graphs of certain kind,we give a formula of the independence polynomial of double-light trees.Moreover,we prove that the independence polynomial has only real zeros,therefore is log-concave and unimodal.In Chapter 4,conclusion.