Some Extremal Results On The Wiener Indices Of Trees And The Bipartite Wiener Vectors Of Trees

The distance theory is an elementary research branch in the graph theory. In this paper,we mainly study the Wiener index in the distance theory. Given a graph G, its Wiener index W(G) is defined as the sum of distances between all unordered pairs of its vertices, namely,W(G) =∑{u,v}?V(G)dG(u, v), where dG(u, v) is the distance between u and v.The theory of Wiener indices of graphs is an important research direction in the distance theory with many applications in the theoretical and network analysis.In this paper, we mainly study the extremal Wiener indices within certain classes of trees. Moreover, we introduce a new concept of the bipartite Wiener vectors of trees and give an application of this concept in theoretical chemistry.The main results of this thesis is as follows;1. Among all trees of order n with exactly one longest path, we characterize the trees with the first up to the fifth smallest Wiener indices(see Chapter 2).2. By using the concept of centroid in the distance theory, we characterize the trees which minimize the Wiener index among all trees of given order that contain a prescribed subtree(see Chapter 3).3. We characterize the trees which minimize the Wiener index among all trees of given order with given number of segments. We also introduce a new concept of the segment sequences and characterize the trees which minimize the Wiener index among all trees of given number with prescribed segment sequence(see Chapter 4).4. According to Lepovi′c-Gutman’s decomposition on the Wiener indices of bipartite graphs, we introduce a new concept of the bipartite Wiener vectors of trees and present an application in theoretical chemistry(see Chapter 5).