Damir Vukičević (Author), Janez Žerovnik (Author)

Abstract

Recently Nikolić, Trinajstič and Randić put forward a novel modification ▫$^mW(G)$▫ of the Wiener index ▫$W(G)$▫, defined as ▫$^mW(G) = \sum_{u,v \in E(G)} n_G(u,v)^{-1} n_G(v,u)^{-1}$▫. This definition was generalized to $^mW(G) = \sum_{u,v \in E(G)} n_G(u,v)^{\lambda} n_G(v,u)^{\lamba}$ by Gutman and the present authors. Another class of modified indices ▫$_{\lambda$}W(G) = \frac{1}{2} \sum_{uv \in E(G)} (v(G)^\lammda - n_G(u,v)^\lambda - n_G(v,u)^\lambda)$▫ is studied here. It is shown that some of main properties of ▫$W(G)$▫, $^mW(G)$ and $^{\lambda}W(G)$ are also properties of $_{\lambda}W(G)$, valid for all values of the parameter ▫$\lambda \ne 0$▫. 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 ▫$\lambda > 1$▫, ▫$_\lambda W(P_n) > _\lambda W(T_n) > \lambda W(S_n)$▫, vhereas for any ▫$\lambda <1$▫, $_\lambda W(P_n) < _\lambda W(T_n) < \lambda W(S_n)$. Thus ▫$_\lambda W(G)$▫ 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 ▫$_\lambda W(G)$▫ then, in the general case, this ordering is different for different ▫$\lambda$▫.

Keywords

matematika;kemijska teorija grafov;Wienerjev indeks;modificiran Wienerjev indeks;mathematics;chemical graph theory;Wiener index;modified Wiener index;

Data

Language: English
Year of publishing:
Typology: 1.01 - Original Scientific Article
Organization: UM FS - Faculty of Mechanical Engineering
Publisher: Slovensko kemijsko društvo
UDC: 519.17:54
COBISS: 9929238 Link will open in a new window
ISSN: 1318-0207
Views: 676
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Other data

Secondary language: Slovenian
Secondary abstract: Nedavno so Nikolić, Trinajstić in Randić predlagali modifikacijo Wienerjevega števila ▫$W(G)$▫, definirano z ▫$^mW(G) = \sum_{uν∈E(G)} n_G(u,ν)^{-1} n_G(u,ν)^{-1}$▫. Invarianto so Gutman in avtorja posplošili na ▫$^λW(G) = \sum_{uν∈E(G)} n_G(u,ν)^λ n_G(u,ν)^λ$▫. Tu obravnavamo posplošitev podobnega tipa, ▫$W_{min,λ}(G) = \sum_{uν∈E(G)}V(G)^λm_G(u,ν)^λ−m_G(u,ν)^{2λ}$▫) in pokažemo, da nekatere pomembne lastnosti ▫$W(G) $▫, ▫$m^W(G)$▫ in ▫$^λW(G)$▫, veljajo tudi za ▫$W_{min,λ}(G)$▫, za večino vrednosti parametra λ. Dokažemo, da za poljubno drevo (povezan acikličen graf) z n točkami ▫$T_n$▫, ki ni pot ▫$P_n$▫ ali zvezda ▫$S_n$▫, velja ▫$W_{min,λ}(Pn) > W_{min,λ}(T_n) > W_{min,λ}(S_n)$▫, za vse λ ≥ 1 in λ < 0. Za te vrednosti parametra je torej ▫$W_{min,λ}(G)$▫ razred topoloških indeksov, ki so lahko uporabni pri obravnavi od razvejanosti odvisnih lastnosti v QSPR in QSAR. Dokažemo tudi, da so vsi novi indeksi različni v naslednjem smislu: če uredimo vsa drevesa glede na ▫$W_{min,λ}(G)$▫ potem za različne vrednosti parametra λ dobimo različne urejenosti.
Secondary keywords: matematika;kemijska teorija grafov;Wienerjev indeks;modificiran Wienerjev indeks;
URN: URN:NBN:SI
Type (COBISS): Scientific work
Pages: str. 272-281
Volume: ǂVol. ǂ52
Issue: ǂno. ǂ3
Chronology: 2005
ID: 10859165