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1.
The Sierpiński product of graphs
Jurij 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
.pdf Full text (526,44 KB)

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Projekt prenove Železniškega muzeja Ljubljana : zaključna projektna naloga
Jurij Vrenko, 2021, undergraduate thesis

Keywords: muzej, Slovenske železnice, projekt, elaborat, železnice, zgodovina, management
Published in RUP: 02.02.2022; Views: 924; Downloads: 59
.pdf Full text (1,97 MB)

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Regular polygonal systems
Jurij Kovič, 2019, original scientific article

Keywords: regular polygonal system, boundary code, face vector, symmetry group, reconstructibility from the boundary
Published in RUP: 03.01.2022; Views: 839; Downloads: 17
.pdf Full text (353,82 KB)

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Classification of convex polyhedra by their rotational orbit Euler characteristic
Jurij 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
.pdf Full text (272,96 KB)

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