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1.
Hereditarnia 2019
2019, other monographs and other completed works

Abstract: The booklet contains the abstracts of the talks given at the 22th Hereditarnia Workshop on Graph Properties that was held at the Faculty of Electrical Engineering and Computer Science in Maribor on 21st and 22nd of June, 2019. The workshop attracted 22 participants from 8 countries. All of the participants are researchers in di˙erent areas of graph theory, but at this event they all presented topics connected with (hereditary) graph properties. Themes of the talks encompass a wide range of contemporary graph theory research, notably, various types of graph colorings, graph domination, some graph dimensions matchings and graph products. Beside the abstracts of the plenary speaker (Roman Sotak) and three invited speakers (Tanja Gologranc, Michael A. Henning and Ismael G. Yero), the booklet also contains the abstracts of 7 contributed talks given at the event.
Keywords: mathematics, graph theory, Hereditarnia, Maribor, Slovenia
Published: 13.12.2019; Views: 279; Downloads: 100
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2.
Ljubljana-Leoben graph theory seminar
2017, other monographs and other completed works

Abstract: The booklet contains the abstracts of the talks given at the 30th Ljubljana-Leoben Graph Theory Seminar that was held at the Faculty of Natural Sciences and Mathematics in Maribor between 13-15 September, 2017. The seminar attracted more than 30 participants from eight countries, all of which are researchers in different areas of graph theory. The topics of the talks encompass a wide range of contemporary graph theory research, notably, various types of graph colorings (b-coloring, packing coloring, edge colorings), graph domination (rainbow domination, Grundy domination, graph security), distinguishing problems, algebraic graph theory, graph algorithms, chemical graph theory, coverings, matchings and also some classical extremal problems. Beside the abstracts of the four invited speakers (Csilla Bujtás, Premysl Holub, Jakub Przybyło, Zsolt Tuza), the booklet contains also the abstracts of 18 contributed talks given at the event.
Keywords: mathematics, discrete mathematics, graph theory, Ljubljana-Leoben seminar, Maribor, Slovenia
Published: 08.12.2017; Views: 554; Downloads: 60
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3.
Omega polynomial revisited
Mircea V. Diudea, Sandi Klavžar, 2010, original scientific article

Abstract: Omega polynomial was proposed by Diudea (Omega Polynomial, Carpath. J. Math., 2006, 22, 43-47) to count the opposite topologically parallel edges in graphs, particularly to describe the polyhedral nanostructures. In this paper, the main definitions are re-analyzed and clear relations with other three related polynomials are established. These relations are supported by close formulas and appropriate examples.
Keywords: mathematics, chemical graph theory, counting polynomials, Ommega polynomial, Theta polynomial, Pi polynomial, PI index, Sadhana polynomial, Cluj-Ilmenau index, CI index
Published: 24.08.2017; Views: 377; Downloads: 65
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4.
Roman domination number of the Cartesian products of paths and cycles
Polona Repolusk, Janez Žerovnik, 2012, original scientific article

Abstract: Roman domination is a historically inspired variety of general domination such that every vertex is labeled with labels from $\{0,1,2\}$. Roman domination number is the smallest of the sums of labels fulfilling condition that every vertex, labeled 0, has a neighbor, labeled 2. Using algebraic approach we give ▫$O(C)$▫ time algorithm for computing Roman domination number of special classes of polygraphs (rota- and fasciagraphs). By implementing the algorithm we give formulas for Roman domination number of the Cartesian products of paths and cycles ▫$P_n \Box P_k$▫, ▫$P_n \Box C_k$▫ for ▫$k \leq 8$▫ and ▫$n \in {\mathbb N}$▫ and for ▫$C_n \Box P_k$▫ and ▫$C_n \Box C_k$▫ for ▫$k \leq 5$▫, ▫$n \in {\mathbb N}$▫. We also give a list of Roman graphs among investigated families.
Keywords: graph theory, Roman domination number, Cartesian product, polygraphs, path algebra
Published: 23.08.2017; Views: 529; Downloads: 140
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5.
Altered Wiener indices
Damir 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: 17.08.2017; Views: 439; Downloads: 75
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6.
The number of moves of the largest disc in shortest paths on Hanoi graphs
Simon Aumann, Katharina Götz, Andreas Hinz, Ciril Petr, 2014, original scientific article

Abstract: In contrast to the widespread interest in the Frame-Stewart conjecture (FSC) about the optimal number of moves in the classical Tower of Hanoi task with more than three pegs, this is the first study of the question of investigating shortest paths in Hanoi graphs ▫$H_p^n$▫ in a more general setting. Here ▫$p$▫ stands for the number of pegs and ▫$n$▫ for the number of discs in the Tower of Hanoi interpretation of these graphs. The analysis depends crucially on the number of largest disc moves (LDMs). The patterns of these LDMs will be coded as binary strings of length ▫$p-1$▫ assigned to each pair of starting and goal states individually. This will be approached both analytically and numerically. The main theoretical achievement is the existence, at least for all ▫$n \geqslant p(p-2)$▫, of optimal paths where ▫$p-1$▫ LDMs are necessary. Numerical results, obtained by an algorithm based on a modified breadth-first search making use of symmetries of the graphs, lead to a couple of conjectures about some cases not covered by our ascertained results. These, in turn, may shed some light on the notoriously open FSC.
Keywords: graph theory, Tower of Hanoi, Hanoi graphs, shortest paths, symmetries, breadth-first search
Published: 14.08.2017; Views: 688; Downloads: 141
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7.
Linear recognition of generalized Fibonacci cubes $Q_h (111)$
Yoomi Rho, Aleksander Vesel, 2016, original scientific article

Abstract: The generalized Fibonacci cube $Q_h(f)$ is the graph obtained from the $h$-cube $Q_h$ by removing all vertices that contain a given binary string $f$ as a substring. In particular, the vertex set of the 3rd order generalized Fibonacci cube $Q_h(111)$ is the set of all binary strings $b_1b_2 ... b_h$ containing no three consecutive 1’s. We present a new characterization of the 3rd order generalized Fibonacci cubes based on their recursive structure. The characterization is the basis for an algorithm which recognizes these graphs in linear time.
Keywords: graph theory, Fibonacci cubes, recognition algorithm
Published: 10.07.2017; Views: 512; Downloads: 82
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8.
Efficient open domination in graph products
Dorota Kuziak, Iztok Peterin, Ismael G. Yero, 2014, original scientific article

Abstract: A graph ▫$G$▫ is an efficient open domination graph if there exists a subset ▫$D$▫ of ▫$V(G)$▫ for which the open neighborhoods centered in vertices of ▫$D$▫ form a partition of ▫$V(G)$▫. We completely describe efficient open domination graphs among lexicographic, strong, and disjunctive products of graphs. For the Cartesian product we give a characterization when one factor is ▫$K_2$▫.
Keywords: graph theory, efficient open domination, graph products, total domination
Published: 10.07.2017; Views: 459; Downloads: 70
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9.
Covering codes in Sierpiński graphs
Laurent Beaudou, Sylvain Gravier, Sandi Klavžar, Matjaž Kovše, Michel Mollard, 2010, original scientific article

Abstract: Za dani graf ▫$G$▫ in celi števili ▫$a$▫ in ▫$b$▫ je ▫$(a,b)$▫-koda grafa ▫$G$▫ množica vozlišč ▫$C$▫, tako da ima vsako vozlišče iz ▫$C$▫ natanko ▫$a$▫ sosedov v ▫$C$▫, vsako drugo vozlišče pa natanko ▫$b$▫ sosedov v ▫$C$▫. V tem prispevku klasificiramo števila ▫$a$▫ in ▫$b$▫, za katera obstajajo ▫$(a,b)$▫-kode v grafih Sierpińskega.
Keywords: graph theory, codes in graphs, perfect codes, Sierpiński graphs
Published: 10.07.2017; Views: 431; Downloads: 81
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10.
A new characterization and a recognition algorithm of Lucas cubes
Andrej Taranenko, 2013, original scientific article

Abstract: Fibonacci and Lucas cubes are induced subgraphs of hypercubes obtained by excluding certain binary strings from the vertex set. They appear as models for interconnection networks, as well as in chemistry. We derive a characterization of Lucas cubes that is based on a peripheral expansion of a unique convex subgraph of an appropriate Fibonacci cube.This serves as the foundation for a recognition algorithm of Lucas cubes that runs in linear time.
Keywords: graph theory, Lucas cubes, characterization, recognition algorithm
Published: 10.07.2017; Views: 564; Downloads: 67
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