Research On The Wiener Index Of Graphs

The Wiener index is the sum of distances for all pairs of vertices with a connected graph.It is an important topological index in Chemistry defined by Harold Wiener in 1947 which is used to describe the structure and properties of molecule. After several years, Mathematician began to pay attention to this index and described it in Mathematical words. In his papers,Wiener did not use the math-theoretical language. Graph Theory can be used to describe the structure of molecule, as a useful mathematical language, so it is a good tool that is for the Wiener index. We will use the standard language of graph theory and study this important index.Firstly, we investigate the properties of the Wiener index of unicyclic graphs,which are used to give a lower bound for Wiener index of unicylic graphs of order 2βhaving perfect matching. Moreover,all extremal unicyclic graphs which attain the lower bound are characterized. And then, we obtain the extremal trees with the second smallest Wiener indices among all the n-vertex trees with perfect matching. Secondly, by using the majorization partial ordering of degree sequences, we characterize the extremal trees with the second minimum Wiener indices among all the n-vertex trees with given matching number or independent number,respectively. Finally, for eachλ≠0,we determine the extremal graphs inΤ( n,Δ)andΤ_1,Δ ( n)with the maximalλ-modified Wiener indices,respectively.