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1.
On non-normal arc-transitive 4-valent dihedrants
István 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.
Najdeno v: ključnih besedah
Povzetek najdenega: ...non-normal 4-valent arc-transitive Cayley graph on a dihedral group ▫$D_n$▫ such that ▫$X$▫ is bipartite,...
Ključne besede: Cayley graph, arc transitivity, dihedral group
Objavljeno: 15.10.2013; Ogledov: 1711; Prenosov: 36
URL Polno besedilo (0,00 KB)

2.
Classification of 2-arc-transitive dihedrants
Shao 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 ▫${\mathbb{F}}_q$▫.
Najdeno v: ključnih besedah
Povzetek najdenega: ...2-arc-transitive dihedrants, that is, Cayley graphs of dihedral groups is given, thus completing the study...
Ključne besede: permutation group, imprimitive group, dihedral group, Cayley graph, dihedrant, 2-Arc-transitive graph
Objavljeno: 15.10.2013; Ogledov: 1524; Prenosov: 44
URL Polno besedilo (0,00 KB)

3.
Distance-regular Cayley graphs on dihedral groups
Štefko Miklavič, Primož Potočnik, 2007, 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.
Najdeno v: ključnih besedah
Povzetek najdenega: ...classification of distance-regular Cayley graphs on dihedral groups. There exist four obvious families of such...
Ključne besede: mathematics, grah theory, distance-regular graph, distance-transitive graph, Cayley graph, dihedral group, dihedrant, difference set
Objavljeno: 15.10.2013; Ogledov: 1209; Prenosov: 52
URL Polno besedilo (0,00 KB)

4.
Distance-regular Cayley graphs on dihedral groups
Primož Potočnik, Štefko Miklavič, 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.
Najdeno v: ključnih besedah
Povzetek najdenega: ...classification of distance-regular Cayley graphs on dihedral groups. There exist four obvious families of such...
Ključne besede: mathematics, grah theory, distance-regular graph, distance-transitive graph, Cayley graph, dihedral group, dihedrant, difference set
Objavljeno: 10.07.2015; Ogledov: 830; Prenosov: 39
URL Polno besedilo (0,00 KB)

5.
Further restrictions on the structure of finite DCI-groups
Edward Dobson, Joy Morris, Pablo Spiga, 2015, izvirni znanstveni članek

Najdeno v: ključnih besedah
Povzetek najdenega: ...isomorphism problem, CI-group, dihedral group, Cayley graph, ...
Ključne besede: isomorphism problem, CI-group, dihedral group, Cayley graph
Objavljeno: 08.08.2016; Ogledov: 891; Prenosov: 22
URL Polno besedilo (0,00 KB)

6.
Which Haar graphs are Cayley graphs?
István Estélyi, Tomaž Pisanski, 2016, izvirni znanstveni članek

Najdeno v: ključnih besedah
Povzetek najdenega: ...Haar graph, Cayley graph, dihedral group, generalized dihedral group, ...
Ključne besede: Haar graph, Cayley graph, dihedral group, generalized dihedral group
Objavljeno: 16.11.2017; Ogledov: 880; Prenosov: 24
URL Polno besedilo (0,00 KB)

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