Research On Three Kinds Of Graph Parameters Based On Resistance Distance

Graph parameters are important research contents in graph theory.Topological indices are graph parameters that quantitatively describe molecular structure,and they are important tools to study molecular graphs.Topological indices are closely related to physical properties,chemical properties and thermodynamic parameters of compounds,so they have strong application value.Based on the resistance distance,the Kirchhoff index is one of the famous graph parameters.The multiplicative degree-Kirchhoff index and the additive degree-Kirchhoff index are two important modifications of the Kirchhoff index.The three kinds of Kirchhoffian graph parameters are not only important invariants of graphs,but also have a wide range of practical applications in physics,chemistry and network.Let G be a finite simple connected graph.The Kirchhoff index of G,the multiplicative degree-Kirchhoff index of G and the additive degree-Kirchhoff index of G are respectively defined as and where dG(u)denotes the degree of the vertex u in G,and RG(u,v)denotes the resistance distance between vertices u and v in G.In this thesis,we mainly study the three kinds of Kirchhoffian graph parameters of unicyclic graphs.This thesis consists of four chapters as follows.In the first chapter,we first give some basic definitions and notation related to this thesis,then introduce the research background,significance and research progress at home and abroad,and finally give the main research results obtained in this thesis.In the second chapter,we mainly use one unified method to determine maximum Kirchhoff index,maximum multiplicative degree-Kirchhoff index and maximum additive degree-Kirchhoff index of n-vertex unicyclic graphs with given maximum degree,and characterize their extremal graphs.Furthermore,maximum Kirchhoff index,maximum multiplicative degree-Kirchhoff index and maximum additive degree-Kirchhoff index of n-vertex unicyclic graphs are determined.In the third chapter,we study the multiplicative degree-Kirchhoff index of unicyclic graphs under the condition of given order.We mainly determine the first seven maximum multiplicative degree-Kirchhoff indices of n-vertex unicyclic graphs,where n≥ 14,and the corresponding graphs whose multiplicative degree-Kirchhoff indices achieve these values.In the fourth chapter,based on the existing research results,we propose some new research directions and subjects.