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21.
Cayley graphs on abelian groups
Edward Dobson, Pablo Spiga, Gabriel Verret, 2015, izvirni znanstveni članek

Ključne besede: Cayley graph, abelian group, automorphism
Objavljeno v RUP: 03.04.2017; Ogledov: 2263; Prenosov: 136
URL Povezava na celotno besedilo

22.
Quotients of CI-Groups are CI-Groups
Edward Dobson, Joy Morris, 2015, izvirni znanstveni članek

Ključne besede: CI-group, Cayley graph, Quotient
Objavljeno v RUP: 03.04.2017; Ogledov: 2065; Prenosov: 139
URL Povezava na celotno besedilo

23.
24.
On colour-preserving automorphisms of Cayley graphs
Klavdija Kutnar, Ademir Hujdurović, Edward Dobson, Dave Witte Morris, Joy Morris, 2016, objavljeni povzetek znanstvenega prispevka na konferenci

Ključne besede: Cayley graph, automorphism, group, colouring
Objavljeno v RUP: 08.08.2016; Ogledov: 2569; Prenosov: 87
URL Povezava na celotno besedilo

25.
On the split structure of lifted groups
Aleksander Malnič, Rok Požar, 2016, izvirni znanstveni članek

Opis: Let ▫$\wp \colon \tilde{X} \to X$▫ be a regular covering projection of connected graphs with the group of covering transformations ▫$\rm{CT}_\wp$▫ being abelian. Assuming that a group of automorphisms ▫$G \le \rm{Aut} X$▫ lifts along $\wp$ to a group ▫$\tilde{G} \le \rm{Aut} \tilde{X}$▫, the problem whether the corresponding exact sequence ▫$\rm{id} \to \rm{CT}_\wp \to \tilde{G} \to G \to \rm{id}$▫ splits is analyzed in detail in terms of a Cayley voltage assignment that reconstructs the projection up to equivalence. In the above combinatorial setting the extension is given only implicitly: neither ▫$\tilde{G}$▫ nor the action ▫$G\to \rm{Aut} \rm{CT}_\wp$▫ nor a 2-cocycle ▫$G \times G \to \rm{CT}_\wp$▫, are given. Explicitly constructing the cover ▫$\tilde{X}$▫ together with ▫$\rm{CT}_\wp$▫ and ▫$\tilde{G}$▫ as permutation groups on ▫$\tilde{X}$▫ is time and space consuming whenever ▫$\rm{CT}_\wp$▫ is large; thus, using the implemented algorithms (for instance, HasComplement in Magma) is far from optimal. Instead, we show that the minimal required information about the action and the 2-cocycle can be effectively decoded directly from voltages (without explicitly constructing the cover and the lifted group); one could then use the standard method by reducing the problem to solving a linear system of equations over the integers. However, along these lines we here take a slightly different approach which even does not require any knowledge of cohomology. Time and space complexity are formally analyzed whenever ▫$\rm{CT}_\wp$▫ is elementary abelian.
Ključne besede: algorithm, abelian cover, Cayley voltages, covering projection, graph, group extension, group presentation, lifting automorphisms, linear systems over the integers, semidirect product
Objavljeno v RUP: 15.10.2015; Ogledov: 2675; Prenosov: 157
.pdf Celotno besedilo (422,56 KB)

26.
27.
Distance-regular Cayley graphs on dihedral groups
Štefko Miklavič, Primož Potočnik, 2005, izvirni znanstveni članek

Opis: The main result of this article is a classification of distance-regular Cayley graphs on dihedral groups. There exist four obvious families of such graphs, which are called trivial. These are: complete graphs, complete bipartite graphs, complete bipartite graphs with the edges of a 1-factor removed, and cycles. It is proved that every non-trivial distance-regular Cayley graph on a dihedral group is bipartite, non-antipodal, has diameter 3 and arises either from a cyclic di#erence set, or possibly (if any such exists) from a dihedral difference set satisfying some additional conditions. Finally, all distance-transitive Cayley graphs on dihedral groups are determined. It transpires that a Cayley graph on a dihedral group is distance-transitive if and only if it is trivial, or isomorphic to the incidence or to the non-incidence graph of a projective space ▫$\mathrm{PG}_{d-1} (d,q)$▫, ▫$d \ge 2$▫, or the unique pair of complementary symmetric designs on 11 vertices.
Ključne besede: mathematics, grah theory, distance-regular graph, distance-transitive graph, Cayley graph, dihedral group, dihedrant, difference set
Objavljeno v RUP: 10.07.2015; Ogledov: 2454; Prenosov: 89
URL Povezava na celotno besedilo

28.
On overgroups of regular abelian p-groups
Edward Dobson, 2009, izvirni znanstveni članek

Opis: Let ▫$G$▫ be a transitive group of odd prime-power degree whose Sylow ▫$p$▫-subgroup ▫$P$▫ is abelian od rank ▫$t$▫. Weshow that if ▫$p > 2^{t-1}$▫, then ▫$G$▫ has a normal subgroup that is a direct product of ▫$t$▫ permutation groups of smaller degree that are either cyclic or doubly-transitive simple groups. As a consequence, we determine the full automorphism group of a Cayley diagraph of an abelian group with rank two such that the Sylow ▫$p$▫-subgroup of the full automorphism group is abelian.
Ključne besede: group theory, graph theory, Cayley graph, abelian group, regular group, p-group
Objavljeno v RUP: 15.10.2013; Ogledov: 3133; Prenosov: 157
URL Povezava na celotno besedilo

29.
30.
Hamiltonian cycles in Cayley graphs whose order has few prime factors
Klavdija Kutnar, Dragan Marušič, D. W. Morris, Joy Morris, Primož Šparl, 2012, izvirni znanstveni članek

Opis: We prove that if Cay▫$(G; S)$▫ is a connected Cayley graph with ▫$n$▫ vertices, and the prime factorization of ▫$n$▫ is very small, then Cay▫$(G; S)$▫ has a hamiltonian cycle. More precisely, if ▫$p$▫, ▫$q$▫, and ▫$r$▫ are distinct primes, then ▫$n$▫ can be of the form kp with ▫$24 \ne k < 32$▫, or of the form ▫$kpq$▫ with ▫$k \le 5$▫, or of the form ▫$pqr$▫, or of the form ▫$kp^2$▫ with ▫$k \le 4$▫, or of the form ▫$kp^3$▫ with ▫$k \le 2$▫.
Ključne besede: graph theory, Cayley graphs, hamiltonian cycles
Objavljeno v RUP: 15.10.2013; Ogledov: 3389; Prenosov: 120
.pdf Celotno besedilo (545,91 KB)

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