On indices of Wiener and anti-Wiener type

Abstract

In this paper we define Wiener and anti-Wiener type of indices, so that we first introduce ordering of tree graphs, and then define that a topological index is of Wiener type if it is an increasing function with respect to the introduced order. Similarly, we define that a topological index is of anti-Wiener type if it is a decreasing function with respect to the introduced order. The introduced order of tree graphs has the star Sn for minimal graph, while the path Pn is the maximal graph. Therefore, all indices of Wiener type obtain minimum value for Sn and maximum value for Pn, while the reverse holds for indices of anti-Wiener type. Then we introduce a simple criterion on edge contribution function of a topological index which enables us to establish if a topological index is of Wiener or anti-Wiener type. Finally, we apply our result to several generalizations of Wiener index, such as modified Wiener indices, variable Wiener indices and Steiner k-Wiener index.