Research On Expected Values Of Several Important Topological Indexes In Random Polyphenylene Chains

Let G be a connected graph,and let i,j be any two vertices in graph G.The distance between i and j is the length of a shortest path connecting them.The resistance distance rij between vertices i,j of a connected graph G is the effective resistance between them.The Wiener index of graph G is defined as the sum of the distances between all pair of unordered vertices in G.In the same way,the Kirchhoff index of graph G is defined as the sum of the resistance distances between all pair of unordered vertices in G.The Wiener index and the Kirchhoff index are important invariants in graph theory.It has a wide range of applications in the study of mathematical chemistry.It is well-known that the weighted version of the Wiener index and the Kirch-hoff index are also important graph parameters.It is better to apply the weighted Wiener index (resp.the weighted Kirchhoff index) to study quantitative structural activity and quantitative structural properties than those of the unweighted version.Therefore,it is very meaningful to study the weighted Wiener index and the weighted Kirchhoff index of graphs.Our thesis mainly studies the weighted Wiener index and the weighted Kirchhoff index of linear random polyphenylene chains,respectively.The concrete content is as follows:·In Chapter 1,we introduce the background and significance of the research,including the development of a representative at home and abroad regarding this aspect.Based on the background and the profound discussion, it fully shows the main work' s necessity.We give our main results in this section.·In Chapter 2,we give some necessary notations and terminologies·In Chapter 3,we give the proofs of Theorem 1.1 and 1.2.We determine firstly the mathematical expectation of Gutman index and Schultz index of linear random polyphenylene chain Gn.Then,we get the maximum and minimum values of Gutman index and Schultz index of Gn in the sense of average value,and the corresponding extremal graphs are characterized.Finally,we determine the average values of Schultz index and Gutman index of linear polyphenylene chain with n hexagons.·In Chapter 4,we give the proofs of Theorem 1.3 and 1.4.firstly,we deter-mine the expected value of multiplicative degree-Kirchhoff and additive degree-Kirchhoff index of linear random polyphenylene chain Gn,respectively.Then,we get the maximum and minimum values of multiplicative degree-Kirchhoff index and additive degree-Kirchhoff index of Gn in the sense of average value,and characterize the polar graph when the maximum value is obtained.At last,we obtain the the average values of multiplicative degree-Kirchhoff index and additive degree-Kirchhoff index of linear polyphenylene chain with n hexagons.·In Chapter 5,we summarize the main results of this thesis and give some further research prospects.