Damir Vukičević^a and Janez Žerovnik^b
a Department of Mathematics, University of Split, Croatia
^b Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, SI-2000 Maribor, Slovenia and IMFM,
Jadranska 19, SI-1000 Ljubljana, Slovenia E-mail: janez.zerovnik@fmf.uni-lj.si

Abstract
Recently Nikolić, Trinajstić and Randić put forward a novel modification ^mW(G) of the Wiener number W(G), called modified Wiener index , which definition was generalized later by Gutman and the present authors. Here we study another class of modified indices defined as W_min,λ(G)=∑(V(G)^λm_G(u,n)^λ-m_G(u,n)^2λ) and show that some of the important properties of W(G), ^mW(G) and ^λW(G)  are also properties of W_min,λ(G), valid for most values of the parameter λ. 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 λ≥1 or λ < 0, W_min,λ(P_n)>W_min,λ(T_n)>W_min,λ(S_n). Thus for these values of the parameter λ, W_min,λ(G) 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 W_min,λ(G) then, in the general case, this ordering is different for different λ.

Key words: Wiener number, modified Wiener indices, branching, chemical graph theory