Altered Wiener Indices

Damir Vukičevića and Janez Žerovnikb
Department of Mathematics, University of Split, Croatia
Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, SI-2000 Maribor, Slovenia and IMFM,
Jadranska 19, SI-1000 Ljubljana, Slovenia E-mail:

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 Wmin,λ(G)=∑(V(G)λmG(u,n)λ-mG(u,n)2λ) and show that some of the important properties of W(G), mW(G) and λW(G)  are also properties of Wmin,λ(G), valid for most values of the parameter λ. In particular, if Tn is any n-vertex tree, different from the n-vertex path Pn and the n-vertex star Sn , then for any λ1 or λ < 0, Wmin,λ(Pn)>Wmin,λ(Tn)>Wmin,λ(Sn). Thus for these values of the parameter λ, Wmin,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 Wmin,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