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A note on acyclic number of planar graphs
Mirko Petruševski, Riste Škrekovski, 2017, izvirni znanstveni članek

Opis: The acyclic number ▫$a(G)$▫ of a graph ▫$G$▫ is the maximum order of an induced forest in ▫$G$▫. The purpose of this short paper is to propose a conjecture that ▫$a(G)\geq \left( 1-\frac{3}{2g}\right)n$▫ holds for every planar graph ▫$G$▫ of girth ▫$g$▫ and order ▫$n$▫, which captures three known conjectures on the topic. In support of this conjecture, we prove a weaker result that ▫$a(G)\geq \left( 1-\frac{3}{g} \right)n$▫ holds. In addition, we give a construction showing that the constant ▫$\frac{3}{2}$▫ from the conjecture cannot be decreased.
Ključne besede: induced forest, acyclic number, planar graph, girth
Objavljeno v RUP: 03.01.2022; Ogledov: 1051; Prenosov: 16
.pdf Celotno besedilo (227,50 KB)

Classification of convex polyhedra by their rotational orbit Euler characteristic
Jurij Kovič, 2017, izvirni znanstveni članek

Opis: Let ▫$\mathcal P$▫ be a polyhedron whose boundary consists of flat polygonal faces on some compact surface ▫$S(\mathcal P)$▫ (not necessarily homeomorphic to the sphere ▫$S^{2}$)▫. Let ▫$vo_{R}(\mathcal P), eo_{R}(\mathcal P)$▫, ▫$ fo_{R}(\mathcal P)$▫ be the numbers of rotational orbits of vertices, edges and faces, respectively, determined by the group ▫$G = G_{R}(P)$▫ of all the rotations of the Euclidean space ▫$E^{3}$▫ preserving ▫$\mathcal P$▫. We define the ''rotational orbit Euler characteristic'' of ▫$\mathcal P$▫ as the number ▫$Eo_{R}(\mathcal P) = vo_{R}(\mathcal P) - eo_{R}(\mathcal P) + fo_{R}(\mathcal P)$▫. Using the Burnside lemma we obtain the lower and the upper bound for ▫$Eo_{R}(\mathcal P)$▫ in terms of the genus of the surface ▫$S(P)$▫. We prove that ▫$Eo_{R} \in \lbrace 2,1,0,-1\rbrace $▫ for any convex polyhedron ▫$\mathcal P$▫. In the non-convex case ▫$Eo_{R}$▫ may be arbitrarily large or small.
Ključne besede: polyhedron, rotational orbit, Euler characteristic
Objavljeno v RUP: 03.01.2022; Ogledov: 900; Prenosov: 18
.pdf Celotno besedilo (272,96 KB)

Vertex-transitive graphs and their arc-types
Marston D. E. Conder, Tomaž Pisanski, Arjana Žitnik, 2017, izvirni znanstveni članek

Opis: Let ▫$X$▫ be a finite vertex-transitive graph of valency ▫$d$▫, and let ▫$A$▫ be the full automorphism group of ▫$X$▫. Then the arc-type of ▫$X$▫ is defined in terms of the sizes of the orbits of the stabiliser ▫$A_v$▫ of a given vertex ▫$v$▫ on the set of arcs incident with ▫$v$▫. Such an orbit is said to be self-paired if it is contained in an orbit ▫$\Delta$▫ of ▫$A$▫ on the set of all arcs of v$X$▫ such that v$\Delta$▫ is closed under arc-reversal. The arc-type of ▫$X$▫ is then the partition of ▫$d$▫ as the sum ▫$n_1 + n_2 + \dots + n_t + (m_1 + m_1) + (m_2 + m_2) + \dots + (m_s + m_s)$▫, where ▫$n_1, n_2, \dots, n_t$▫ are the sizes of the self-paired orbits, and ▫$m_1,m_1, m_2,m_2, \dots, m_s,m_s$▫ are the sizes of the non-self-paired orbits, in descending order. In this paper, we find the arc-types of several families of graphs. Also we show that the arc-type of a Cartesian product of two "relatively prime" graphs is the natural sum of their arc-types. Then using these observations, we show that with the exception of ▫$1+1$▫ and ▫$(1+1)$▫, every partition as defined above is \emph{realisable}, in the sense that there exists at least one vertex-transitive graph with the given partition as its arc-type.
Ključne besede: symmetry type, vertex-transitive graph, arc-transitive graph, Cayley graph, cartesian product, covering graph
Objavljeno v RUP: 03.01.2022; Ogledov: 924; Prenosov: 19
.pdf Celotno besedilo (475,17 KB)

Relative edge betweenness centrality
Damir Vukičević, Riste Škrekovski, Aleksandra Tepeh, 2017, izvirni znanstveni članek

Opis: We introduce a new edge centrality measure - relative edge betweenness ▫$\gamma (uv) = b(uv)/\sqrt{c(u)c(v)}$▫, where ▫$b(uv$)▫ is the standard edge betweenness and ▫$c(u)$▫ is the adjusted vertex betweenness. In this alternative definition, the importance of an edge is normalized with respect to the importance of its end-vertices. This gives a better presentation of the ''local'' importance of an edge, i.e. its importance in the near neighborhood. We present sharp upper and lower bounds on this invariant together with the characterization of graphs attaining these bounds. In addition, we discuss the bounds for various interesting graph families, and state several open problems.
Objavljeno v RUP: 03.01.2022; Ogledov: 780; Prenosov: 15
.pdf Celotno besedilo (261,26 KB)

A decomposition for Markov processes at an independent exponential time
Mihael Perman, 2017, izvirni znanstveni članek

Opis: The path of Markov process ▫$X$▫ run up to an independent exponential random time ▫$S_\theta$▫ can be decomposed into the part prior to the last exit time from a point before ▫$S_\theta$▫, and the remainder up to ▫$S_\theta$▫. In this paper the laws of the two segments are identified under suitable assumptions using excursion theory.
Objavljeno v RUP: 03.01.2022; Ogledov: 752; Prenosov: 13
.pdf Celotno besedilo (312,49 KB)

A novel characterization of cubic Hamiltonian graphs via the associated quartic graphs
Simona Bonvicini, Tomaž Pisanski, 2017, izvirni znanstveni članek

Opis: We give a necessary and sufficient condition for a cubic graph to be Hamiltonian by analyzing Eulerian tours in certain spanning subgraphs of the quartic graph associated with the cubic graph by 1-factor contraction. This correspondence is most useful in the case when it induces a blue and red 2-factorization of the associated quartic graph. We use this condition to characterize the Hamiltonian ▫$I$▫-graphs, a further generalization of generalized Petersen graphs. The characterization of Hamiltonian ▫$I$▫-graphs follows from the fact that one can choose a 1-factor in any ▫$I$▫-graph in such a way that the corresponding associated quartic graph is a graph bundle having a cycle graph as base graph and a fiber and the fundamental factorization of graph bundles playing the role of blue and red factorization. The techniques that we develop allow us to represent Cayley multigraphs of degree 4, that are associated to abelian groups, as graph bundles. Moreover, we can find a family of connected cubic (multi)graphs that contains the family of connected ▫$I$▫-graphs as a subfamily.
Ključne besede: generalized Petersen graphs, I-graphs, Hamiltonian cycles, Eulerian tours, Cayley multigraphs
Objavljeno v RUP: 03.01.2022; Ogledov: 866; Prenosov: 16
.pdf Celotno besedilo (1,01 MB)

On [plus/minus] 1 eigenvectors of graphs
Dragan Stevanović, 2016, izvirni znanstveni članek

Opis: While discussing his spectral bound on the independence number of a graph, Herbert Wilf asked back in 1986 what kind of a graph admits an eigenvector consisting solely of ▫$\pm 1$▫ entries? We prove that Wilf's problem is NP-complete, but also that the set of graphs having a ▫$\pm 1$▫ eigenvector is quite rich, being closed under a number of different graph compositions.
Ključne besede: eigenvector, adjacency matrix, Wilf's problem
Objavljeno v RUP: 03.01.2022; Ogledov: 825; Prenosov: 28
.pdf Celotno besedilo (325,02 KB)

Mathematical aspects of fullerenes
Vesna Andova, František Kardoš, Riste Škrekovski, 2016, izvirni znanstveni članek

Opis: Fullerene graphs are cubic, 3-connected, planar graphs with exactly 12 pentagonal faces, while all other faces are hexagons. Fullerene graphs are mathematical models of fullerene molecules, i.e., molecules comprised only by carbon atoms different than graphites and diamonds. We give a survey on fullerene graphs from our perspective, which could be also considered as an introduction to this topic. Different types of fullerene graphs are considered, their symmetries, and construction methods. We give an overview of some graph invariants that can possibly correlate with the fullerene molecule stability, such as: the bipartite edge frustration, the independence number, the saturation number, the number of perfect matchings, etc.
Ključne besede: fullerene, cubic graph, planar graph, topological indices
Objavljeno v RUP: 03.01.2022; Ogledov: 839; Prenosov: 17
.pdf Celotno besedilo (626,25 KB)

Mathematical aspects of Wiener index
Martin Knor, Riste Škrekovski, Aleksandra Tepeh, 2016, izvirni znanstveni članek

Opis: The Wiener index (i.e., the total distance or the transmission number), defined as the sum of distances between all unordered pairs of vertices in a graph, is one of the most popular molecular descriptors. In this article we summarize some results, conjectures and problems on this molecular descriptor, with emphasis on works we were involved in.
Ključne besede: Wiener index, total distance, topological index, molecular descriptor, chemical graph theory
Objavljeno v RUP: 03.01.2022; Ogledov: 1227; Prenosov: 38
.pdf Celotno besedilo (434,58 KB)

On colour-preserving automorphisms of Cayley graphs
Ademir Hujdurović, Klavdija Kutnar, Dave Witte Morris, Joy Morris, 2016, izvirni znanstveni članek

Opis: We study the automorphisms of a Cayley graph that preserve its natural edge-colouring. More precisely, we are interested in groups ▫$G$▫, such that every such automorphism of every connected Cayley graph on ▫$G$▫ has a very simple form: the composition of a left-translation and a group automorphism. We find classes of groups that have the property, and we determine the orders of all groups that do not have the property. We also have analogous results for automorphisms that permute the colours, rather than preserving them.
Ključne besede: Cayley graph, automorphism, colour-preserving, colour-permuting
Objavljeno v RUP: 03.01.2022; Ogledov: 849; Prenosov: 19
.pdf Celotno besedilo (412,93 KB)

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