1. The Sierpiński product of graphsJurij Kovič, Tomaž Pisanski, Sara Sabrina Zemljič, Arjana Žitnik, 2023, original scientific article Abstract: In this paper we introduce a product-like operation that generalizes the construction of the generalized Sierpiński graphs. Let ▫$G, \, H$▫ be graphs and let ▫$f: V(G) \to V(H)$▫ be a function. Then the Sierpiński product of graphs ▫$G$▫ and ▫$H$▫ with respect to ▫$f$▫, denoted by ▫$G\otimes_f H$▫, is defined as the graph on the vertex set ▫$V(G) \times V(H)$▫, consisting of ▫$|V(G)|$▫ copies of ▫$H$▫; for every edge ▫$\{g, g'\}$▫ of ▫$G▫$ there is an edge between copies ▫$gH$▫ and ▫$g'H$▫ of form ▫$\{(g, f(g'), (g', f(g))\}$▫. Some basic properties of the Sierpiński product are presented. In particular, we show that the graph ▫$G\otimes_f H$▫ is connected if and only if both graphs ▫$G$▫ and ▫$H$▫ are connected and we present some conditions that ▫$G, \, H$▫ must fulfill for ▫$G\otimes_f H$▫ to be planar. As for symmetry properties, we show which automorphisms of ▫$G$▫ and ▫$H$▫ extend to automorphisms of ▫$G\otimes_f H$▫. In several cases we can also describe the whole automorphism group of the graph ▫$G\otimes_f H$▫. Finally, we show how to extend the Sierpiński product to multiple factors in a natural way. By applying this operation ▫$n$▫ times to the same graph we obtain an alternative approach to the well-known ▫$n$▫-th generalized Sierpiński graph.
Keywords: Sierpiński graphs, graph products, connectivity, planarity, symmetry
Published in RUP: 06.11.2023; Views: 314; Downloads: 3
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3. Combinatorial configurations, quasiline arrangements, and systems of curves on surfacesJürgen Bokowski, Jurij Kovič, Tomaž Pisanski, Arjana Žitnik, 2018, original scientific article Keywords: pseudoline arrangement, quasiline arrangement, projective plane, incidence structure, combinatorial configuration, topological configuration, geometric configuration, sweep, wiring diagram, allowable sequence of permutations, maps on surfaces
Published in RUP: 03.01.2022; Views: 753; Downloads: 17
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4. Classification of convex polyhedra by their rotational orbit Euler characteristicJurij Kovič, 2017, original scientific article Abstract: 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.
Keywords: polyhedron, rotational orbit, Euler characteristic
Published in RUP: 03.01.2022; Views: 796; Downloads: 18
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5. Point-ellipse configurations and related topicsGábor Gévay, Nino Bašić, Jurij Kovič, Tomaž Pisanski, 2021, original scientific article Keywords: point-line configuration, conic section, point-ellipse configuration, point-conic configuration, Levi graph, Carnot's theorem
Published in RUP: 18.10.2021; Views: 1254; Downloads: 21
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9. Uporaba simetrijskih grafov pri konveksnih poliedrihJurij Kovič, 2013, doctoral dissertation Abstract: Grafe praporov in simetrijske grafe, ki so jih odkrili okrog leta 1980, so doslej uporabljali predvsem za klasifikacijo zemljevidov - to je grafov, celično vloženih v kompaktne ploskve. Namen te disertacije je prikazati nekaj razširitev njihove uporabe. V prvem delu disertacije uporabimo simetrijske grafe pri klasifikaciji poliedrov z regularnimi poligonskimi ali zvezdastimi lici. Podamo tudi karakterizacijo teh poliedrov z minimalnim številom parametrov. V drugem delu disertacije na podoben način klasificiramo molekule,sestavljene iz pravilnih šestkotniških gradnikov, pri čemer upoštevamo tudi njihove točkovne grupe. V tretjem delu disertacije pojem simetrijskih grafov razširimo na hiperzemljevide in geometrijske konfiguracije, nazadnje pa tudi na sferne poliedre in sferne molekule.
Keywords: polieder, graf praporov, simetrijski graf, simetrijska grupa
Published in RUP: 15.10.2013; Views: 2906; Downloads: 56
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