The Study On The Kirchhoff Indices And The Corresponding Extremal Problem In (n,n+2)-graphs

Let G be a connected graph,we view G as an electrical network N by con-sidering each edge of G as a unit resistor.The resistance distance between two vertices of G,is defined to be the effective resistance between two nodes as com-puted with Ohm’s law in N.The Kirchhoff index of G,is the sum of resistance distance between all pairs of vertices in G.The study about invariant of graph is one of the important directions of graph theory.The resistance distance and the Kirchhoff index are the invariants of the molecular structure drawing.They are introduced formally by Klein and Randic in 1993.We assume that the num-ber of vertices of G is n,the（n,n+2）-graphs are the connected graphs with cyclomatic number 2.For the connected graphs whose cyclomatic number less than two,their resistance distances and the Kirchhoff indices have been described well.In this paper,we discuss the（n,n+2）-graphs mainly.The path of G with the degrees of end-vertices not less than 3 is called the internal path of G.By the number of the internal paths of G,the（n,n+2）-graphs are divided into four kinds,such as the fist kind does not contain internal path,the second kind contains an internal path,the third kind contains two internal paths and the fourth kind contains three internal paths.Furthermore,we can transform one（n,n+2）-graph to another in each kind according the changes of the lengths of the circle and internal path,or the locations of root vertices in hanging trees.In this thesis,with some graph transformations and the algebraic or analytical methods,we study the maximal and the minimal Kirchhoff indices of（n,n+2）-graphs by fixing the lengths of circles and the number of internal paths in each kind,and vary the locations of root vertices in hanging trees.Assume that the length of each circle is p,the two tapes of graphs in the first kind are denoted byτ_n^3pandτ_n^3*prespectively,and the two tapes of graphs in the second kind are denoted byτ_n^2p,pandτ_n^2*p,prespectively,and the two tapes of graphs in the third kind are denoted byτ_n^p,p,pandτ_n^p,2prespectively,the fourth kind is denoted byτ_n^p*p*p.The main results as follows:1.In the chapter two,we simplifyτ_n^3pandτ_n^3*p,and obtain the maximal and the minimal Kirchhoff indices and the corresponding extremal graphs.2.In the chapter three,we simplifyτ_n^2p,pandτ_n^2*p,p,and obtain the maximal and the minimal Kirchhoff indices and the corresponding extremal graphs.3.In the chapter four,we simplifyτ_n^p,p,pandτ_n^p,2p,and obtain the maximal and the minimal Kirchhoff indices and the corresponding extremal graphs.4.In the chapter five,we simplifyτ_n^p*p*p,and obtain the maximal and the minimal Kirchhoff indices and the corresponding extremal graphs.