engleski

A Class of Modified Wiener Indices

The Wiener index of a tree T obeys the relation W(T) = Sigma(e) n(1)(e) (.) n(2)(e) where n(1)(e) and n(2)(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 Nikolic, Trinajstic and Randic put forward a novel modification W-m of the Wiener index, defined as W-m(T) = Sigma(e) [n(1)(e) (.) n(2)(e)](-1). We now extend their definition as W-m(lambda)(T) = Sigma(e) [n(1)(e) (.) n(2)(e)](lambda), and show that some of the main properties of both W and W-m are, in fact, properties of W-m(lambda), valid for all values of the parameter lambda not equal 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 lambda, W-m(lambda)(P-n) > W-m(lambda)(T-n) > W-m(lambda)(S-n), whereas for any negative lambda, W-m(lambda)(P-n) < W-m(lambda)(T-n) < W-m(lambda)(S-n). Thus W-m(lambda) 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-m(lambda) then, in the general case, this ordering is different for different lambda

Wiener index; Modified wiener index; Nikolic-trinajstic-randic index branching; Chemical graph theory

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