**Altered Wiener Indices****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*
*^{m}W(*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*)^{λ}m_{G}(*u*,n)^{λ}-*m*_{G}(*u*,n)^{2}^{λ})
and show that some of the important properties of *W*(*G*), ^{
m}W(*G*) and
^{λ}W(*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