1. Altered Wiener indicesDamir Vukičević, Janez Žerovnik, 2005, original scientific article Abstract: 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.
Keywords: mathematics, chemical graph theory, Wiener index, modified Wiener index
Published in DKUM: 17.08.2017; Views: 1227; Downloads: 124
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2. Simplified computation of matchings in polygraphsAnte Graovac, Damir Vukičević, Damir Ježek, Janez Žerovnik, 2005, original scientific article Abstract: Matching polynomial and perfect matchings for fasciagraphs, rotagraphs and twisted rotagraphs are treated in the paper. Classical transfer matrix approach makes it possible to get recursions for matching polynomial and perfect matchings, but the order of the matrix grows exponentially in the number of the linking edges between monographs. Novel transfer matrices are introduced whose order is much lower than that in classical transfer matrices. The virtue of the method introduced is especially pronounced when twoor more linking edges end in the same terminal vertex of a monograph. An example of a polyacene polygraph with extended pairings is given where a novel matrix has only 16 entries as compared to 65536 entries in the classical transfer matrix. However, all pairings are treated here on equal footing, but the method introduced can be applied to selected types of pairings of interest in chemistry.
Keywords: polygraphs, matching polynomial, matchings, perfect matchings, Kekulé structures, extended structures, recursive enumeration, transfer matrix method
Published in DKUM: 05.07.2017; Views: 1583; Downloads: 92
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3. On algebraic and geometric Kekulé structures in benzenoid rotagraphsAnte Graovac, Damir Vukičević, Janez Žerovnik, 2006, original scientific article Abstract: Recently introduced algebraic Kekulé structures (AKS) describe the ▫$\pi$▫-electron distribution within rings of a conjugated network. The ratio of the AKS countto the classical Kekulé structures count was studied in benzenoid rotagraphs. By considering three representative classes of such rotagraphs, it was shown that this ratio tends towards either 1 or 0, or its value lies between 0 and 1.
Keywords: Kekulé structures, Kekulé structure count, geometric and algebraic Kekulé structures, benzenoids, rotagraph
Published in DKUM: 05.07.2017; Views: 1230; Downloads: 101
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4. A class of modified Wiener indicesIvan Gutman, Damir Vukičević, Janez Žerovnik, 2004, original scientific article 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: graph theory, chemical graph theory, modified Wiener index, Nikolić-Trinajstić-Randić index, branching
Published in DKUM: 05.07.2017; Views: 1196; Downloads: 100
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5. Binary coding of algebraic Kekulé structures of catacondensed benzenoid graphsDamir Vukičević, Petra Žigert Pleteršek, 2008, original scientific article Abstract: Algebraična Kekuléjeva struktura končnega katakondenziranega benzenoidnega grafa s ▫$h$▫ šestkotniki je podana z binarno kodo dolžine ▫$h$▫. Postopek je obrnljiv in sicer lahko iz binarne kode rekonstruiramo algebraično Kekuléjevo strukturo.
Keywords: matematika, kemijska teorija grafov, benzenoidni ogljikovodiki, benzenoidni grafi, Kekuléjeve strukture, Randićeve strukture, 1-faktor, binarno kodiranje, mathematics, chemical graph theory, benzenoid hydrocarbons, benzenoid graph, Kekulé structures, algebraic Kekulé structures, Randić structures, 1-factor, binary coding
Published in DKUM: 10.07.2015; Views: 1256; Downloads: 99
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