31. On quartic half-arc-transitive metacirculantsDragan Marušič, Primož Šparl, 2008, izvirni znanstveni članek Opis: Following Alspach and Parsons, a metacirculant graph is a graph admitting a transitive group generated by two automorphisms ▫$\rho$▫ and ▫$\sigma$▫, where ▫$\rho$▫ is ▫$(m,n)$▫-semiregular for some integers ▫$m \ge 1$▫, ▫$n \ge 2▫$, and where ▫$\sigma$▫ normalizes ▫$\rho$▫, cyclically permuting the orbits of ▫$\rho$▫ in such a way that ▫$\sigma^m$▫ has at least one fixed vertex. A half-arc-transitive graph is a vertex- and edge- but not arc-transitive graph. In this article quartic half-arc-transitive metacirculants are explored and their connection to the so called tightly attached quartic half-arc-transitive graphs is explored. It is shown that there are three essentially different possibilities for a quartic half-arc-transitive metacirculant which is not tightly attached to exist. These graphs are extensively studied and some infinite families of such graphs are constructed.
Ključne besede: mathematics, graph theory, metacirculant graph, half-arc-transitive graph, tightly attached, automorphism group
Objavljeno v RUP: 15.10.2013; Ogledov: 4269; Prenosov: 133
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32. Classification of 2-arc-transitive dihedrantsShao Fei Du, Aleksander Malnič, 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 ▫${\mathbb{F}}_q$▫.
Ključne besede: permutation group, imprimitive group, dihedral group, Cayley graph, dihedrant, 2-Arc-transitive graph
Objavljeno v RUP: 15.10.2013; Ogledov: 3762; Prenosov: 91
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33. On non-normal arc-transitive 4-valent dihedrantsIstván Kovács, Boštjan Kuzman, Aleksander Malnič, 2010, izvirni znanstveni članek Opis: Let ▫$X$▫ be a connected non-normal 4-valent arc-transitive Cayley graph on a dihedral group ▫$D_n$▫ such that ▫$X$▫ is bipartite, with the two bipartition sets being the two orbits of the cyclic subgroup within ▫$D_n$▫. It is shown that ▫$X$▫ is isomorphic either to the lexicographic product ▫$C_n[2K_1]$▫ with ▫$n \geq 4$▫ even, or to one of the five sporadic graphs on 10, 14, 26, 28 and 30 vertices, respectively.
Ključne besede: Cayley graph, arc transitivity, dihedral group
Objavljeno v RUP: 15.10.2013; Ogledov: 4153; Prenosov: 123
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36. Vertex-transitive expansions of (1, 3)-treesMarko Lovrečič Saražin, Dragan Marušič, 2010, objavljeni znanstveni prispevek na konferenci Opis: A nonidentity automorphism of a graph is said to be semiregular if all of its orbits are of the same length. Given a graph ▫$X$▫ with a semiregular automorphism ▫$\gamma$▫, the quotient of ▫$X$▫ relative to ▫$\gamma$▫ is the multigraph ▫$X/\gamma$▫ whose vertices are the orbits of ▫$\gamma$▫ and two vertices are adjacent by an edge with multiplicity ▫$r$▫ if every vertex of one orbit is adjacent to ▫$r$▫ vertices of the other orbit. We say that ▫$X$▫ is an expansion of ▫$X/\gamma$▫. In [J.D. Horton, I.Z. Bouwer, Symmetric ▫$Y$▫-graphs and ▫$H$▫-graphs, J. Combin. Theory Ser. B 53 (1991) 114-129], Hortonand Bouwer considered a restricted sort of expansions (which we will call :strong" in this paper) where every leaf of ▫$X/\gamma$▫ expands to a single cycle in ▫$X$▫. They determined all cubic arc-transitive strong expansions of simple ▫$(1,3)$▫-trees, that is, trees with all of their vertice shaving valency 1 or 3, thus extending the classical result of Frucht, Graver and Watkins (see [R. Frucht, J.E. Graver, M.E. Watkins, The groups of the generalized Petersen graphs, Proc. Cambridge Philos. Soc. 70 (1971) 211-218]) about arc-transitive strong expansions of ▫$K_2$▫ (also known as the generalized Petersen graphs). In this paper another step is taken further by considering the possible structure of cubic vertex-transitive expansions of general ▫$(1,3)$▫-multitrees (where vertices with double edges are also allowed); thus the restriction on every leaf to be expanded to a single cycle is dropped.
Ključne besede: graph, tree, cubic, vertex-transitive, arc-transitive, expansion
Objavljeno v RUP: 15.10.2013; Ogledov: 4281; Prenosov: 81
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