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1.
Classification of 2-arc-transitive dihedrants
Shao Fei Du, Aleksander Malnič, Dragan Marušič, 2008, original scientific article

Abstract: 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 ▫${\mathbb{F}}_q$▫.
Found in: ključnih besedah
Summary of found: ... permutation group, imprimitive group, dihedral group, Cayley graph,... ...dihedrants, that is, Cayley graphs of dihedral groups is given, thus completing the study of...
Keywords: permutation group, imprimitive group, dihedral group, Cayley graph, dihedrant, 2-Arc-transitive graph
Published: 15.10.2013; Views: 1551; Downloads: 46
URL Full text (0,00 KB)

2.
Primitive bicirculant association schemes and a generalization of Wielandt's theorem
István Kovács, Dragan Marušič, Mikhail Muzychuk, 2010, original scientific article

Found in: ključnih besedah
Summary of found: ... permutation group, association scheme, primitive association scheme, ...
Keywords: permutation group, association scheme, primitive association scheme
Published: 15.10.2013; Views: 1562; Downloads: 63
URL Full text (0,00 KB)

3.
Decomposition of skew-morphisms of cyclic groups
István Kovács, Roman Nedela, 2011, original scientific article

Abstract: 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 ▫$(n, \varphi(n)) = 1$▫.
Found in: ključnih besedah
Summary of found: ...skew-morphism of a group ▫$H$▫ is a permutation ▫$\sigma$▫ of its elements fixing the identity...
Keywords: cyclic group, permutation group, skew-morphism, Schur ring
Published: 15.10.2013; Views: 1880; Downloads: 51
URL Full text (0,00 KB)

4.
Distance-transitive graphs admit semiregular automorphisms
Klavdija Kutnar, Primož Šparl, 2010, original scientific article

Abstract: 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.
Found in: ključnih besedah
Summary of found: ...to ▫$v'$▫. A semiregular element of a permutation group is anon-identity element having all cycles...
Keywords: distance-transitive graph, vertex-transitive graph, semiregular automorphism, permutation group
Published: 15.10.2013; Views: 1644; Downloads: 52
URL Full text (0,00 KB)

5.
Minimal normal subgroups of transitive permutation groups of square-free degree
Aleksander Malnič, Dragan Marušič, Edward Dobson, 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]).
Found in: ključnih besedah
Summary of found: ...It is shown that a minimal normal sub group of a transitive permutation group of square-free...
Keywords: mathematics, graph theory, transitive permutation group, 2-closed group, square-free degree, semiregular automorphism, vertex-transitive graph
Published: 03.04.2017; Views: 855; Downloads: 45
URL Full text (0,00 KB)

6.
Semiregular automorphisms of vertex-transitive graphs of certain valencies
Dragan Marušič, Edward Dobson, Aleksander Malnič, 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]).
Found in: ključnih besedah
Summary of found: ...and, more generally, that all 2-closed transitive permutation groups contain such an element (see [D.... ...prime, admitting a transitive action of a ▫$\{2,p\}$▫- group, has a non-identity semiregular automorphism. As a...
Keywords: mathematics, graph theory, transitive permutation group, 2-closed group, semiregular automorphism, vertex-transitive graph
Published: 03.04.2017; Views: 817; Downloads: 41
URL Full text (0,00 KB)

7.
Multiply transitive permutation groups via the small Mathieu groups
Anna Klymenko, 2012, master's thesis

Found in: ključnih besedah
Summary of found: ...permutation group, group action, orbit, stabilizer, multiply transitive group,...
Keywords: permutation group, group action, orbit, stabilizer, multiply transitive group, affine group, Mathieu group
Published: 13.11.2017; Views: 346; Downloads: 7
URL Full text (0,00 KB)

8.
Odd automorphisms in vertex-transitive graphs
Ademir Hujdurović, Klavdija Kutnar, Dragan Marušič, 2016, original scientific article

Abstract: Avtomorfizem grafa se imenuje sod/lih, če deluje na množici vozlišč kot soda/liha permutacija. V tem članku zastavimo problem določitve tistih točkovno-tranzitivnih grafov, ki premorejo lihe avtomorfizme. Predstavimo delne rezultate za določene razrede točkovno-tranzitivnih grafov, med drugim za Cayleyjeve grafe. Kot posledico teh rezultatov dobimo karakterizacijo ločno-tranzitivnih cirkulantov brez lihih avtomorfizmov.
Found in: ključnih besedah
Summary of found: ...graph, vertex-transitive, automorphism group, even permutation, odd permutation, ...
Keywords: graph, vertex-transitive, automorphism group, even permutation, odd permutation
Published: 15.11.2017; Views: 562; Downloads: 42
URL Full text (0,00 KB)

9.
Odd extensions of transitive groups via symmetric graphs - The cubic case
Klavdija Kutnar, Dragan Marušič, 2018, original scientific article

Abstract: When dealing with symmetry properties of mathematical objects, one of the fundamental questions is to determine their full automorphism group. In this paper this question is considered in the context of even/odd permutations dichotomy. More precisely: when is it that the existence of automorphisms acting as even permutations on the vertex set of a graph, called even automorphisms, forces the existence of automorphisms that act as odd permutations, called odd automorphisms. As a first step towards resolving the above question, complete information on the existence of odd automorphisms in cubic symmetric graphs is given.
Found in: ključnih besedah
Summary of found: ...automorphism group, arc-transitive, even permutation, odd permutation, cubic symmetric graph... ...questions is to determine their full automorphism group. In this paper this question is considered...
Keywords: automorphism group, arc-transitive, even permutation, odd permutation, cubic symmetric graph
Published: 19.11.2018; Views: 258; Downloads: 73
URL Full text (0,00 KB)

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