Original scientific paper

A Class of Modified Wiener Indices

Ivan Gutman ; Faculty of Science, University of Kragujevac, 34000 Kragujevac, Serbia and Montenegro
Damir Vukičević ; Department of Mathematics, University of Split, Croatia
Janez Žerovnik ; Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, SI-2000 Maribor, Slovenia

Abstract

The Wiener index of a tree T obeys the relation W(T) = Σ_en₁(e) • n₂(e) where n₁(e) and n₂(e) are the number of vertices on the two sides of the edge e, and where the summation goes over all edges of T. Recently Nikolić, Trinajstić and Randić put forward a novel modification mW of the Wiener index, defined as mW(T) = Σ_e[n₁(e) • n₂(e)]–1. We now extend their definition as mW_λ(T) = Σ_e[n₁(e) • n₂(e)]λ, and show that some of the main properties of both W and mW are, in fact, properties of mW_λ, valid for all values of the parameter λ≠0. In particular, if T_n is any n-vertex tree, different from the n-vertex path P_n and the n-vertex star S_n, then for any positive λ, mW_λ(P_n) > mW_λ(T_n) > mW_λ(S_n), whereas for any negative λ, mW_λ(P_n) < mW_λ(T_n) < mW_λ(S_n). Thus mW_λ provides a novel class of structure-descriptors, suitable for modeling branching-dependent properties of organic compounds, applicable in QSPR and QSAR studies. We also demonstrate that if trees are ordered with regard to mW_λ then, in the general case, this ordering is different for different λ.

Keywords

Wiener index; modified Wiener index; Nikolić-Trinajstić-Randić index; branching; chemical graph theory

Hrčak ID:

102652

URI

https://hrcak.srce.hr/102652

Publication date:

31.5.2004.

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