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1.
Computing quadratic entropy in evolutionary trees
Matt DeVos, Arne Ø. Mooers, Drago Bokal, Sandi Klavžar, Aki Mimoto, 2011

Opis: We note here that quadratic entropy, a measure of biological diversity introduced by Rao, is a variant of the weighted Wiener index, a graph invariant intensively studied in mathematical chemistry. This fact allows us to deduce some efficient algorithms for computing the quadratic entropy in the case of given tip weights, which may be useful for community biodiversity measures. Furthermore, on ultrametric phylogenetic trees, the maximum of quadratic entropy is a measure of pairwise evolutionary distinctness in conservation biology, introduced by Pavoine. We present an algorithm that maximizes this quantity in linear time, offering a significant improvement over the currently used quadratic programming approaches.
Ključne besede: teorija grafov, evolucijsko drevo, filogenetsko drevo, Wienerjev indeks, graph theory, evolutionary tree, phylogenetic tree, quadratic entropy, originality, distinctness, Wiener index
Objavljeno: 10.07.2015; Ogledov: 239; Prenosov: 4
URL Celotno besedilo (0,00 KB)

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On the canonical metric representation, average distance, and partial Hamming graphs
Sandi Klavžar, 2006, izvirni znanstveni članek

Opis: Povprečna razdalja grafa je izražena s pomočjo kanonične metrične reprezentacije. Enakost lahko preoblikujemo v neenakost tako, da karakterizira izometrične podgrafe Hammingovih grafov. Ta pristop poenostavlja prepoznavanje teh grafov ter izračun povprečne razdalje.
Ključne besede: matematika, teorija grafov, kanonična metrična reprezentacija, Hammingovi grafi, delni Hammingovi grafi, Wienerjev indeks, algoritem prepoznavanja, mathematics, graph theory, cononical metric representation, Hamming graphs, partial Hamming graphs, Wiener index, recognition algorithm
Objavljeno: 10.07.2015; Ogledov: 248; Prenosov: 7
URL Celotno besedilo (0,00 KB)

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Computing quadratic entropy in evolutionary trees
Arne Ø. Mooers, Aki Mimoto, Drago Bokal, Matt DeVos, Sandi Klavžar, 2011, izvirni znanstveni članek

Opis: Kvadratična entropija, ki jo je vpeljal Rao, je mera za biološko raznolikost. V članku opazimo, da je kvadratična entropija inačica uteženega Wienerjevega indeksa, ki je po drugi strani intenzivno raziskovana grafovska invarianta v matematični kemiji. To dejstvo omogoča izpeljavo nekaj učinkovitih algoritmov za izračunavanje kvadratične entropije v primeru danih listnih uteži. Na ultrametričnih drevesih je Pavoine vpeljal maksimum kvadratičnih entropij kot mero za paroma evolucijsko različnost v ohranitveni biologiji. Predstavljamo algoritem, ki maksimizira to količino v linearnem času, kar je pomembna izboljšava glede na obstoječe kvadratične programske pristope.
Ključne besede: teorija grafov, evolucijsko drevo, filogenetsko drevo, kvadratična entropija, različnost, Wienerjev indeks, graph theory, evolutionary tree, phylogenetic tree, quadratic entropy, originality, distinctness, Wiener index
Objavljeno: 10.07.2015; Ogledov: 195; Prenosov: 4
URL Celotno besedilo (0,00 KB)

6.
A class of modified Wiener indices
Ivan 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: 05.07.2017; Ogledov: 107; Prenosov: 3
.pdf Celotno besedilo (125,08 KB)

7.
Corroborating a modification of the Wiener index
Ivan Gutman, Janez Žerovnik, 2002, kratki znanstveni prispevek

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: 05.07.2017; Ogledov: 85; Prenosov: 7
.pdf Celotno besedilo (85,05 KB)

8.
Altered Wiener indices
Damir 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: 17.08.2017; Ogledov: 77; Prenosov: 14
.pdf Celotno besedilo (991,46 KB)

9.
Wiener index of strong product of graphs
Iztok Peterin, Petra Žigert, 2018, izvirni znanstveni članek

Opis: The Wiener index of a connected graph ▫$G$▫ is the sum of distances between all pairs of vertices of ▫$G$▫. The strong product is one of the four most investigated graph products. In this paper the general formula for the Wiener index of the strong product of connected graphs is given. The formula can be simplified if both factors are graphs with the constant eccentricity. Consequently, closed formulas for the Wiener index of the strong product of a connected graph ▫$G$▫ with a cycle are derived.
Ključne besede: Wiener index, graph product, strong product
Objavljeno: 30.11.2017; Ogledov: 157; Prenosov: 4
.pdf Celotno besedilo (424,67 KB)

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