1. Načrtovanje matematičnih dejavnosti : diplomska nalogaKatarina Tratar, 2020, undergraduate thesis Keywords: matematika, načrtovanje, matematična področja, prikriti kurikulum, matematične dejavnosti, mathematics, planning, mathematical fields, covert curriculum, mathematical activities Published in RUP: 22.07.2020; Views: 1638; Downloads: 110 Full text (519,09 KB) |
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3. Minimal normal subgroups of transitive permutation groups of square-free degreeEdward Dobson, Aleksander Malnič, Dragan Marušič, Lewis A. Nowitz, 2007, original scientific article Abstract: It is shown that a minimal normal subgroup of a transitive permutation group of square-free degree in its induced action is simple and quasiprimitive, with three exceptions related to ▫$A_5$▫, ▫$A_7$▫, and PSL(2,29). Moreover, it is shown that a minimal normal subgroup of a 2-closed permutation group of square-free degree in its induced action is simple. As an almost immediate consequence, it follows that a 2-closed transitive permutation group of square-free degree contains a semiregular element of prime order, thus giving a partial affirmative answer to the conjecture that all 2-closed transitive permutation groups contain such an element (see [D. Marušic, On vertex symmetric digraphs,Discrete Math. 36 (1981) 69-81; P.J. Cameron (Ed.), Problems from the fifteenth British combinatorial conference, Discrete Math. 167/168 (1997) 605-615]). Keywords: mathematics, graph theory, transitive permutation group, 2-closed group, square-free degree, semiregular automorphism, vertex-transitive graph Published in RUP: 03.04.2017; Views: 2496; Downloads: 89 Link to full text |
4. Symmetry structure of bicirculantsAleksander Malnič, Dragan Marušič, Primož Šparl, Boštjan Frelih, 2007, original scientific article Abstract: An ▫$n$▫-bicirculant is a graph having an automorphism with two orbits of length ▫$n$▫ and no other orbits. Symmetry properties of ▫$p$▫-bicirculants, ▫$p$▫ a prime, are extensively studied. In particular, the actions of their automorphism groups are described in detail in terms of certain algebraic representation of such graphs. Keywords: mathematics, graph theory, graph, circulant, bicirculant, automorphism group Published in RUP: 03.04.2017; Views: 2619; Downloads: 95 Link to full text |
5. On strongly regular bicirculantsAleksander Malnič, Dragan Marušič, Primož Šparl, 2007, original scientific article Abstract: An ▫$n$▫-bicirculantis a graph having an automorphism with two orbits of length ▫$n$▫ and no other orbits. This article deals with strongly regular bicirculants. It is known that for a nontrivial strongly regular ▫$n$▫-bicirculant, ▫$n$▫ odd, there exists a positive integer m such that ▫$n=2m^2+2m+1▫$. Only three nontrivial examples have been known previously, namely, for ▫$m=1,2$▫ and 4. Case ▫$m=1$▫ gives rise to the Petersen graph and its complement, while the graphs arising from cases ▫$m=2$▫ and ▫$m=4$▫ are associated with certain Steiner systems. Similarly, if ▫$n$▫ is even, then ▫$n=2m^2$▫ for some ▫$m \ge 2$▫. Apart from a pair of complementary strongly regular 8-bicirculants, no other example seems to be known. A necessary condition for the existence of a strongly regular vertex-transitive ▫$p$▫-bicirculant, ▫$p$▫ a prime, is obtained here. In addition, three new strongly regular bicirculants having 50, 82 and 122 vertices corresponding, respectively, to ▫$m=3,4$▫ and 5 above, are presented. These graphs are not associated with any Steiner system, and together with their complements form the first known pairs of complementary strongly regular bicirculants which are vertex-transitive but not edge-transitive. Keywords: mathematics, graph theory, graph, circulant, bicirculant, automorphism group Published in RUP: 03.04.2017; Views: 3879; Downloads: 88 Link to full text |
6. Semiregular automorphisms of vertex-transitive graphs of certain valenciesEdward Dobson, Aleksander Malnič, Dragan Marušič, Lewis A. Nowitz, 2007, original scientific article Abstract: It is shown that a vertex-transitive graph of valency ▫$p+1$▫, ▫$p$▫ a prime, admitting a transitive action of a ▫$\{2,p\}$▫-group, has a non-identity semiregular automorphism. As a consequence, it is proved that a quartic vertex-transitive graph has a non-identity semiregular automorphism, thus giving a partial affirmative answer to the conjecture that all vertex-transitive graphs have such an automorphism and, more generally, that all 2-closed transitive permutation groups contain such an element (see [D. Marušic, On vertex symmetric digraphs, Discrete Math. 36 (1981) 69-81; P.J. Cameron (Ed.), Problems from the Fifteenth British Combinatorial Conference, Discrete Math. 167/168 (1997) 605-615]). Keywords: mathematics, graph theory, transitive permutation group, 2-closed group, semiregular automorphism, vertex-transitive graph Published in RUP: 03.04.2017; Views: 2564; Downloads: 83 Link to full text |
7. Semisymmetric elementary abelian covers of the Möbius-Kantor graphAleksander Malnič, Dragan Marušič, Štefko Miklavič, Primož Potočnik, 2007, original scientific article Abstract: Let ▫$\wp_N : \tilde{X} \to X$▫ be a regular covering projection of connected graphs with the group of covering transformations isomorphic to ▫$N$▫. If ▫$N$▫ is an elementary abelian ▫$p$▫-group, then the projection ▫$\wp_N$▫ is called ▫$p$▫-elementary abelian. The projection ▫$\wp_N$▫ is vertex-transitive (edge-transitive) if some vertex-transitive (edge-transitive) subgroup of Aut ▫$X$▫ lifts along ▫$\wp_N$▫, and semisymmetric if it is edge- but not vertex-transitive. The projection ▫$\wp_N$▫ is minimal semisymmetric if ▫$\wp_N$▫ cannot be written as a composition ▫$\wp_N = \wp \circ \wp_M$▫ of two (nontrivial) regular covering projections, where ▫$\pw_M$▫ is semisymmetric. Finding elementary abelian covering projections can be grasped combinatorially via a linear representation of automorphisms acting on the first homology group of the graph. The method essentially reduces to finding invariant subspaces of matrix groups over prime fields (see [A. Malnic, D. Marušic, P. Potocnik, Elementary abelian covers of graphs, J. Algebraic Combin. 20 (2004) 71-97]). In this paper, all pairwise nonisomorphic minimal semisymmetric elementary abelian regular covering projections of the Möbius-Kantor graph, the Generalized Petersen graph GP(8,3), are constructed. No such covers exist for ▫$p=2$▫. Otherwise, the number of such covering projections is equal to ▫$(p-1)/4$▫ and ▫$1+(p-1)/4$▫ in cases ▫$p \equiv 5,9,13,17,21 \pmod{24}$▫ and ▫$p \equiv 1 \pmod{24}$▫, respectively, and to ▫$(p+1)/4$▫ and ▫$1+(p+1)/4$▫ in cases ▫$p \equiv 3,7,11,15,23 \pmod{24}$▫ and ▫$p \equiv 19 \pmod{24}$▫, respectively. For each such covering projection the voltage rules generating the corresponding covers are displayed explicitly. Keywords: mathematics, graph theory, graph, covering projection, lifting automorphisms, homology group, group representation, matrix group, invariant subspaces Published in RUP: 03.04.2017; Views: 2512; Downloads: 86 Link to full text |
8. Reflexivity defect of spaces of linear operatorsJanko Bračič, Bojan Kuzma, 2009, original scientific article Abstract: For a finite-dimensional linear subspace ▫{$\mathscr{S}} \subseteq {\mathscr{L}} (V,W)$▫ and a positive integer ▫$k$▫, the ▫$k$▫-reflexivity defect of ▫$\mathscr{S}$▫ is defined by ▫${\mathrm{rd}}_k ({\mathscr{S}}) = \dim({\mathrm{Ref}}_k (\mathscr{S})/\mathscr{S})$▫ where ▫${\mathrm{Ref}}_k$▫ is the ▫$k$▫-reflexive closure of ▫$\mathscr{S}$▫. We study this quantity for two-dimensional spaces of operators and for single generated algebras and their commutants. Keywords: mathematics, operator theory, reflexivity defect, reflexivity, two-dimensional space of operators, single generated algebra, commutant Published in RUP: 03.04.2017; Views: 2223; Downloads: 193 Link to full text |
9. Additive rank-one nonincreasing maps on Hermitian matrices over the field GF(2[sup]2)Marko Orel, Bojan Kuzma, 2009, original scientific article Abstract: A complete classification of additive rank-one nonincreasing maps on hermitian matrices over Galois field ▫$GF(2^2)$▫ is obtained. This field is special and was not covered in a previous paper. As a consequence, some known applications, like the classification of additive rank-additivity preserving maps, are extended to arbitrary fields. An application concerning the preservers of hermitian varieties is also presented. Keywords: mathematics, linear algebra, additive preserver, hermitian matrices, rank, Galois field, weak homomorphism of a graph Published in RUP: 03.04.2017; Views: 2622; Downloads: 87 Link to full text |
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