1. Rank-permutable additive mappingsAleksandr Èmilevič Guterman, Anna A. Alieva, Bojan Kuzma, 2006, izvirni znanstveni članek Opis: Let ▫$\sigma$▫ be a fixed non-identical permutation on ▫$k$▫ elements. Additive bijections ▫$T$▫ on the matrix algebra ▫$M_n(\mathbb{F})$▫ over a field ▫$\mathbb{F}$▫ of characteristic zero, with the property that ▫$\rm{rk} (A_1...A_k) = \rm{rk} (A_{\sigma(1)}...A_{\sigma(k)})$▫ implies the same condition on the ▫$T$▫ images, are characterized. It is also shown that the surjectivity assumption can be relaxed, if this property is preserved in both directions. Najdeno v: ključnih besedah Povzetek najdenega: ...Let ▫$\sigma$▫ be a fixed non-identical permutation on ▫$k$▫ elements. Additive bijections ▫$T$▫ on... Ključne besede: mathematics, linearna algebra, matrix algebra, rank, permutation, additive preservers Objavljeno: 15.10.2013; Ogledov: 2632; Prenosov: 85
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2. Classification of 2-arc-transitive dihedrantsAleksander Malnič, Shao Fei Du, Dragan Marušič, 2008, izvirni znanstveni članek Opis: A complete classification of 2-arc-transitive dihedrants, that is, Cayley graphs of dihedral groups is given, thus completing the study of these graphs initiated by the third author in [D. Marušič, On 2-arc-transitivity of Cayley graphs, J. Combin. Theory Ser. B 87 (2003) 162-196]. The list consists of the following graphs: (i) cycles ▫$C_{2n},\; n \ge 3$▫; (ii) complete graphs ▫$K_{2n}, \; n \ge 3$▫; (iii) complete bipartite graphs ▫$K_{n,n}, \; n \ge 3$▫; (iv) complete bipartite graphs minus a matching ▫$K_{n,n} - nK_2, \; n \ge 3$▫; (v) incidence and nonincidence graphs ▫$B(H_{11})$▫ and ▫$B'(H_{11})$▫ of the Hadamard design on 11 points; (vi) incidence and nonincidence graphs ▫$B(PG(d,q))$▫ and ▫$B'(PG(d,q))$▫, with ▫$d \ge 2$▫ and ▫$q$▫ a prime power, of projective spaces; (vii) and an infinite family of regular ▫${\mathbb{Z}}_d$▫-covers ▫$K_{q+1}^{2d}$▫ of ▫$K_{q+1, q+1} - (q+1)K_2$▫, where ▫$q \ge 3$▫ is an odd prime power and ▫$d$▫ is a divisor of ▫$\frac{q-1}{2}$▫ and ▫$q-1$▫, respectively, depending on whether ▫$q \equiv 1 \pmod{4}$▫ or ▫$q \equiv 3 \pmod{4}$▫ obtained by identifying the vertex set of the base graph with two copies of the projective line ▫$PG(1,q)$▫, where the missing matching consists of all pairs of the form ▫$[i,i']$▫, ▫$i \in PG(1,q)$▫, and the edge ▫$[i,j']$▫ carries trivial voltage if ▫$i=\infty$▫ or ▫$j=\infty$▫, and carries voltage ▫$\bar{h} \in {\mathbb{Z}}_d$▫, the residue class of ▫$h \in {\mathbb{Z}}_d$▫, if and only if ▫$i-j = \theta^h$▫, where ▫$\theta$▫ generates the multiplicative group ▫${\mathbb{F}}_q^\ast$▫ of the Galois field ▫${\mathbb{F}}_q$▫. Najdeno v: ključnih besedah Povzetek najdenega: ...permutation group, imprimitive group, dihedral group, Cayley graph,... Ključne besede: permutation group, imprimitive group, dihedral group, Cayley graph, dihedrant, 2-Arc-transitive graph Objavljeno: 15.10.2013; Ogledov: 2773; Prenosov: 85
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3. Primitive bicirculant association schemes and a generalization of Wielandt's theoremDragan Marušič, István Kovács, Mikhail Muzychuk, 2010, izvirni znanstveni članek Najdeno v: ključnih besedah Povzetek najdenega: ...permutation group, association scheme, primitive association scheme, ... Ključne besede: permutation group, association scheme, primitive association scheme Objavljeno: 15.10.2013; Ogledov: 2612; Prenosov: 123
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4. Decomposition of skew-morphisms of cyclic groupsIstván Kovács, Roman Nedela, 2011, izvirni znanstveni članek Opis: A skew-morphism of a group ▫$H$▫ is a permutation ▫$\sigma$▫ of its elements fixing the identity such that for every ▫$x, y \in H$▫ there exists an integer ▫$k$▫ such that ▫$\sigma (xy) = \sigma (x)\sigma k(y)$▫. It follows that group automorphisms are particular skew-morphisms. Skew-morphisms appear naturally in investigations of maps on surfaces with high degree of symmetry, namely, they are closely related to regular Cayley maps and to regular embeddings of the complete bipartite graphs. The aim of this paper is to investigate skew-morphisms of cyclic groups in the context of the associated Schur rings. We prove the following decomposition theorem about skew-morphisms of cyclic groups ▫$\mathbb Z_n$▫: if ▫$n = n_{1}n_{2}$▫ such that ▫$(n_{1}n_{2}) = 1$▫, and ▫$(n_{1}, \varphi (n_{2})) = (\varphi (n_{1}), n_{2}) = 1$▫ (▫$\varphi$▫ denotes Euler's function) then all skew-morphisms ▫$\sigma$▫ of ▫$\mathbb Z_n$▫ are obtained as ▫$\sigma = \sigma_1 \times \sigma_2$▫, where ▫$\sigma_i$▫ are skew-morphisms of ▫$\mathbb Z_{n_i}, \; i = 1, 2$▫. As a consequence we obtain the following result: All skew-morphisms of ▫$\mathbb Z_n$▫ are automorphisms of ▫$\mathbb Z_n$▫ if and only if ▫$n = 4$▫ or ▫$(n, \varphi(n)) = 1$▫. Najdeno v: ključnih besedah Povzetek najdenega: ...skew-morphism of a group ▫$H$▫ is a permutation ▫$\sigma$▫ of its elements fixing the identity... Ključne besede: cyclic group, permutation group, skew-morphism, Schur ring Objavljeno: 15.10.2013; Ogledov: 3100; Prenosov: 104
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5. On cryptographically significant mappings over GF(2 [sup] n)Enes Pašalić, 2008, objavljeni znanstveni prispevek na konferenci Opis: In this paper we investigate the algebraic properties of important cryptographic primitives called substitution boxes (S-boxes). An S-box is a mapping that takes ▫$n$▫ binary inputs whose image is a binary ▫$m$▫-tuple; therefore it is represented as ▫$F:\text{GF}(2)^n \rightarrow \text{GF}(2)^m$▫. One of the most important cryptographic applications is the case ▫$n = m$▫, thus the S-box may be viewed as a function over ▫$\text{GF}(2^n)$▫. We show that certain classes of functions over ▫$\text{GF}(2^n)$▫ do not possess a cryptographic property known as APN (AlmostPerfect Nonlinear) permutations. On the other hand, when ▫$n$▫ is odd, an infinite class of APN permutations may be derived in a recursive manner, that is starting with a specific APN permutation on ▫$\text{GF}(2^k), k$▫ odd, APN permutations are derived over ▫$\text{GF}(2^{k+2i})$▫ for any ▫$i \geq 1$▫. Some theoretical results related to permutation polynomials and algebraic properties of the functions in the ring ▫$\text{GF}(q)[x,y]$▫ are also presented. For sparse polynomials over the field ▫$\text{GF}(2^n)$▫, an efficient algorithm for finding low degree I/O equations is proposed. Najdeno v: ključnih besedah Povzetek najdenega: ...cryptographic property known as APN (AlmostPerfect Nonlinear) permutations. On the other hand, when ▫$n$▫ is... Ključne besede: cryptoanalysis, cryptography, permutation polynomials, power mappings, APN functions, S-box, CCZ-equivalence, algebraic properties Objavljeno: 15.10.2013; Ogledov: 2485; Prenosov: 68
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6. Distance-transitive graphs admit semiregular automorphismsKlavdija Kutnar, Primož Šparl, 2010, izvirni znanstveni članek Opis: A distance-transitive graph is a graph in which for every two ordered pairs ofvertices ▫$(u,v)$▫ and ▫$(u',v')$▫ such that the distance between ▫$u$▫ and ▫$v$▫ is equal to the distance between ▫$u'$▫ and ▫$v'$▫ there exists an automorphism of the graph mapping ▫$u$▫ to ▫$u'$▫ and ▫$v$▫ to ▫$v'$▫. A semiregular element of a permutation group is anon-identity element having all cycles of equal length in its cycle decomposition. It is shown that every distance-transitive graph admits a semiregular automorphism. Najdeno v: ključnih besedah Povzetek najdenega: ...to ▫$v'$▫. A semiregular element of a permutation group is anon-identity element having all cycles... Ključne besede: distance-transitive graph, vertex-transitive graph, semiregular automorphism, permutation group Objavljeno: 15.10.2013; Ogledov: 2716; Prenosov: 93
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7. Cubic symmetric graphs via odd automorphisms, 60th Birthday Lecture Series, Department of Mathematics, University of Auckland, New Zealand, 10 September 2015Klavdija Kutnar, 2015, predavanje na tuji univerzi Najdeno v: ključnih besedah Povzetek najdenega: ...cubic graph, symmetric, automorphism, odd permutation, ... Ključne besede: cubic graph, symmetric, automorphism, odd permutation Objavljeno: 15.10.2015; Ogledov: 1944; Prenosov: 20
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8. Minimal normal subgroups of transitive permutation groups of square-free degreeAleksander Malnič, Edward Dobson, Dragan Marušič, Lewis A. Nowitz, 2007, izvirni znanstveni članek Opis: 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]). Najdeno v: ključnih besedah Povzetek najdenega: Zadetek v naslovu Ključne besede: mathematics, graph theory, transitive permutation group, 2-closed group, square-free degree, semiregular automorphism, vertex-transitive graph Objavljeno: 03.04.2017; Ogledov: 1827; Prenosov: 85
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9. Semiregular automorphisms of vertex-transitive graphs of certain valenciesAleksander Malnič, Edward Dobson, Dragan Marušič, Lewis A. Nowitz, 2007, izvirni znanstveni članek Opis: 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]). Najdeno v: ključnih besedah Povzetek najdenega: ...and, more generally, that all 2-closed transitive permutation groups contain such an element (see [D.... Ključne besede: mathematics, graph theory, transitive permutation group, 2-closed group, semiregular automorphism, vertex-transitive graph Objavljeno: 03.04.2017; Ogledov: 1884; Prenosov: 78
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10. Monomial isomorphisms of cyclic codesEdward Dobson, 2015, izvirni znanstveni članek Najdeno v: ključnih besedah Povzetek najdenega: ...Cyclic code, Monomial, Permutation, Isomorphism, Equivalent, ... Ključne besede: Cyclic code, Monomial, Permutation, Isomorphism, Equivalent Objavljeno: 03.04.2017; Ogledov: 1899; Prenosov: 129
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