Tutte Polynomials For Two Kinds Of Hexagonal Systems

In 1954,Tutte first introduced a bivariate dichromatic canonical form to solve the problem of graph coloring.This dichromatic canonical form was later called the Tutte polynomial,which can reflect some characteristics and properties of the graph.Tutte polynomials are closely related to many graph parameters,and also have close relationship with many subjects,such as physics,chemistry,biology,computer,etc.Therefore,the Tutte polynomial is widely used and is also an important tool for solving other graph parameters.Hexagonal lattice system is the natural graph representation of benzene hydrocar-bons in theoretical chemistry.A hexagonal system graph is a finite and 2 connected planar graph,and each inner surface is surrounded by a hexagon with a side length of 1.In this paper,we use the transfer matrix method to solve the tutte polynomials of two kinds of hexagonal systems,In the first chapter,we first introduce the background and significance of Tutte polynomials,then introduce some concepts and terms relat-ed to this paper,as well as the theorems related to Tutte polynomials.In the second chapter,we first calculate the Tutte polynomial of the catacondensed hexagonal system with only one branch hexagon,then extend it to the catacondensed hexagonal system with multiple branch hexagons,calculate its Tutte polynomial,then solve the number of spanning trees of the graph.In Chapter 3 we first calculate the Tutte polynomials of a hexagonal system with only one interior point and one bifurcation,then extend it to a hexagonal system with k interior points and k bifurcation,and calculate the Tutte polynomials of this system.Finally,the number of spanning trees and its chromatic and stream polynomials are solved.In Chapter 4,the roots of chromatic and stream polynomials in the previous two chapters are studied.