Wiener Index

The Wiener index of a graph is just the sum of distances between all unordered pairs of vertices of the graph. This concept, introduced by the chemist Wiener, is a quite successful tool for designing quantitative structure property relations in organic chemistry, which is closed related to another concept-mean distance, which denoted the average value of the distances between all unordered pairs of vertices in graph. It is used as a measure of the efficiency of a communication network. In a word, these two concepts as important parameters of a graph have been paid attention and further studied by many experts. Based on earlier research on these two concepts, this thesis gives some researches.In Chapter 1, after introducing some important topological indices and basic terminology and notations, we give a brief overview to the main results of the thesis.In Chapter 2, we characterize all trees with the smallest, the second and third smallest values of the Wiener index, and the trees with the largest, the second and third largest values of this index.In Chapter 3, we characterize all unicyclic graphs with the smallest, the second and third smallest values of the Wiener index, and the unicyclic graphs with the largest, the second and third largest values of this index.In Chapter 4, we study the questions concerns the effect of the addition of an edge on the Wiener index of a graph.In Chapter 5, we propose some problems for further research on the Wiener index of tree, the extremal values of Wiener index, and how the Wiener index will change if we change some edges or vertices from a graph.