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On indices of Wiener and anti-Wiener type

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 S-n for minimal graph, while the path P-n is the maximal graph. Therefore, all indices of Wiener type obtain minimum value for S-n and maximum value for P-n 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.

Tree graph ; Order ; Monotonous functions ; Extremal graphs ; Topological index ; Wiener index

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