Ivan Gutman (Author), Damir Vukičević (Author), Janez Žerovnik (Author)

Abstract

The Wiener index of a tree T obeys the relation W(T) = Σen1(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) = Σe[n1(e) • n2(e)]–1. We now extend their definition as mWλ(T) = Σe[n1(e) • n2(e)]λ, and show that some of the main properties of both W and mW are, in fact, properties of mWλ, valid for all values of the parameter λ≠0. 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 positive λ, mWλ(Pn) > mWλ(Tn) > mWλ(Sn), whereas for any negative λ, mWλ(Pn) < mWλ(Tn) < mWλ(Sn). Thus mWλ 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 mWλ then, in the general case, this ordering is different for different λ.

Keywords

teorija grafov;kemijjska teorija grafov;modificiran Wienerjev indeks;indeks Nikolić-Trinajstić-Randić;razvejanost;graph theory;chemical graph theory;modified Wiener index;Nikolić-Trinajstić-Randić index;branching;

Data

Language: English
Year of publishing:
Typology: 1.01 - Original Scientific Article
Organization: UM FS - Faculty of Mechanical Engineering
UDC: 519.17
COBISS: 9803798 Link will open in a new window
ISSN: 0011-1643
Views: 776
Downloads: 85
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Other data

Secondary language: Croatian
Secondary title: Jedna klasa modificiranih Wienerovih indeksa
Secondary abstract: Wienerov indeks stabla T zadovoljava relaciju W(T) = ∑e n1(e) ·n2(e) gdje su n1(e) i n2(e) broj čvorova na dvije strane grane e, i gdje sumiranje ide preko svih grana stabla T. Nedavno su Nikolić, Trinajstić i Randić predložili modifikaciju mW Wienerovog indeksa, definiranu kao mW(T) = ∑e [ n1(e) ·n2(e) ]^–1. Mi sada proširujemo njihovu definiciju na mW λ (T) = ∑e [ n1(e) ·n2(e) ]^λ, i pokazujemo da neka od važnijih svojstava kako W tako i mW važe za mW λ, za svaku vrijednost parametra λ ≠ 0. Ako je Tn bilo koje stablo s n čvorova, različito od puta Pn i zvijezde Sn, onda za svaku pozitivnu λ, mW λ (Pn) > mW λ (Tn) > mW λ (Sn), dok za svaku negativnu λ, mW λ (Pn) < mWß(Tn) < mW λ (Sn). Na taj način mW λ predstavlja novu klasu strukturnih deskriptora, pogodnih za modeliranje o razgranatosti ovisnih svojstava organskih spojeva i učinkovitih u QSPR i QSAR studijama. Također pokazujemo da ako su stabla uređena prema mW λ, tada se, u općem slučaju, ovaj uređaj razlikuje za različite λ.
Secondary keywords: teorija grafov;kemijska teorija grafov;modificiran Wienerjev indeks;indeks Nikolić-Trinajstić-Randić;razvejanost;
URN: URN:SI:UM:
Type (COBISS): Scientific work
Pages: str. 103-109
Volume: ǂVol. ǂ77
Issue: 1/2
Chronology: 2004
ID: 10846955