1. The Graovac-Pisanski index of connected bipartite graphs with applications to hydrocarbon moleculesMatevž Črepnjak, Martin Knor, Niko Tratnik, Petra Žigert Pleteršek, 2021, izvirni znanstveni članek Opis: The Graovac-Pisanski index, also called the modified Wiener index, was introduced in 1991 and represents an extension of the original Wiener index, because it considers beside the distances in a graph also its symmetries. Similarly as Wiener in 1947 showed the correlation of the Wiener indices of the alkane series with the boiling points, in 2018 the connection between the GraovacPisanski index and the melting points of some hydrocarbon molecules was established. In this paper, we prove that the Graovac-Pisanski index of any connected bipartite graph as well as of any connected graph on an even number of vertices is an integer number. These results are applied to some important families of hydrocarbon molecules. By using a computer programme, the graphs with a non-integer Graovac-Pisanski index on at most nine vertices are counted. Finally, an infinite class of unicyclic graphs with a non-integer Graovac-Pisanski index is described.
Ključne besede: modified Wiener index, Graovac-Pisanski index, graph distance, automorphism group, hydrocarbons, carbon nanostructures
Objavljeno v DKUM: 14.02.2025; Ogledov: 0; Prenosov: 4
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2. Altered Wiener indicesDamir Vukičević, Janez Žerovnik, 2005, izvirni znanstveni članek Opis: 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 ▫$W_{min,λ}(G) = ∑(V(G)^λm_G(u,ν)^λ−m_G(u,ν)^{2λ})$▫ and show that some of the important properties of ▫$W(G)$▫, ▫$^mW(G)$▫ and ▫$^λW(G)$▫ 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.
Ključne besede: mathematics, chemical graph theory, Wiener index, modified Wiener index
Objavljeno v DKUM: 17.08.2017; Ogledov: 1227; Prenosov: 126
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3. Corroborating a modification of the Wiener indexIvan Gutman, Janez Žerovnik, 2002, drugi znanstveni članki Opis: In a recent work [Chem. Phys. Lett. 333 (2001) 319-321] Nikolić, Trinajstić, and Randie put forward a novel modification ▫$^m$▫W of the Wiener index. We now show that ▫$^m$▫W possesses the basic properties required by a topological index to be acceptable as a measure of the extent of branching of the carbon-atom skeleton of the respective molecule (and therefore to be a structure-descriptor, potentially applicable in QSPR and QSAR studies). 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 mw(Pn) < mW(Tn) < mW(Sn). We also show how the concept of the modified Wiener index can be extended to weighted molecular graphs.
Ključne besede: graph theory, distance, molecular graphs, modified Wiener index, weigted modified Wiener index, branching, chemical graph theory
Objavljeno v DKUM: 05.07.2017; Ogledov: 1146; Prenosov: 87
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4. A class of modified Wiener indicesIvan Gutman, Damir Vukičević, Janez Žerovnik, 2004, izvirni znanstveni članek Opis: 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.
Ključne besede: graph theory, chemical graph theory, modified Wiener index, Nikolić-Trinajstić-Randić index, branching
Objavljeno v DKUM: 05.07.2017; Ogledov: 1196; Prenosov: 100
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