Constructions for arc-transitive digraphs

Conder, Marston; Lorimer, Peter; Praeger, Cheryl E.

doi:10.1017/s1446788700038477

Cited by 16 publications

(14 citation statements)

References 6 publications

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“…Theorem 3.1 is very powerful and makes it relatively easy to construct many examples of compatible transitive groups. For example, one of the main results of [3] is the construction of a G-arc-transitive digraph Γ with G v ∼ = Alt (6) and (Γ, G) having inlocal and out-local actions the two inequivalent transitive actions of Alt(6) of degree 6. Since, in both of these actions, the point-stabiliser is isomorphic to Alt(5), the existence of such a digraph follows immediately from Theorem 3.1.…”

Section: A Characterisation Of Compatible Transitive Groups and Some mentioning

confidence: 99%

“…Since, in both of these actions, the point-stabiliser is isomorphic to Alt(5), the existence of such a digraph follows immediately from Theorem 3.1. (In fact, infinitely many examples are constructed in [3] but, given one example, one can always construct infinitely many others using standard covering techniques. )…”

Section: A Characterisation Of Compatible Transitive Groups and Some mentioning

confidence: 99%

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Arc-transitive digraphs with quasiprimitive local actions

Giudici

Glasby

et al. 2019

Journal of Pure and Applied Algebra

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Let Γ be a finite G-vertex-transitive digraph. The in-local action of (Γ, G) is the permutation group L − induced by the vertex-stabiliser on the set of in-neighbours of v. The out-local action L + is defined analogously. Note that L − and L + may not be isomorphic. We thus consider the problem of determining which pairs (L − , L + ) are possible. We prove some general results, but pay special attention to the case when L − and L + are both quasiprimitive. (Recall that a permutation group is quasiprimitive if each of its nontrivial normal subgroups is transitive.) Along the way, we prove a structural result about pairs of finite quasiprimitive groups of the same degree, one being (abstractly) isomorphic to a proper quotient of the other.

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Section: A Characterisation Of Compatible Transitive Groups and Some mentioning

confidence: 99%

Section: A Characterisation Of Compatible Transitive Groups and Some mentioning

confidence: 99%

Arc-transitive digraphs with quasiprimitive local actions

Giudici

Glasby

et al. 2019

Journal of Pure and Applied Algebra

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“…The digraphs D 1 (Γ m ) and D 2 (Γ m ) both admit Aut(Γ m ) ∼ = A 2 m+6 as a group of automorphisms, which is quasiprimitive on the vertex set. A systematic study of s-arctransitive digraphs admitting a quasiprimitive group of automorphisms was initiated in [22], and examples of such digraphs for arbitrary large s were first constructed in [9].…”

Section: Tetravalent Edge-transitive Cayley Graphs On Nonabelian Simp...mentioning

confidence: 99%

Tetravalent half-arc-transitive graphs with unbounded nonabelian vertex stabilizers

Xia

2019

Preprint

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Half-arc-transitive graphs are a fascinating topic which connects graph theory, Riemann surfaces and group theory. Although fruitful results have been obtained over the last half a century, it is still challenging to construct half-arc-transitive graphs with prescribed vertex stabilizers. Until recently, there have been only six known connected tetravalent half-arc-transitive graphs with nonabelian vertex stabilizers, and the question whether there exists a connected tetravalent half-arc-transitive graph with nonabelian vertex stabilizer of order 2 s for every s 3 has been wide open. This question is answered in the affirmative in this paper via the construction of a connected tetravalent half-arc-transitive graph with vertex stabilizer D 2 8 × C m 2 for each integer m 1, where D 2 8 is the direct product of two copies of the dihedral group of order 8 and C m 2 is the direct product of m copies of the cyclic group of order 2. The graphs constructed have surprisingly many significant properties in various contexts.

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“…There exist several constructions of finite k-arc-transitive digraphs for arbitrarily large k (see e.g. [3,14,10]). If the digraphs in consideration are infinite, then they might even be highly-arc-transitive, which means that the automorphism group is k-arc-transitive for all k ≥ 0, where 0-arc-transitivity means that the digraph is vertex-transitive.…”

Section: Introductionmentioning

confidence: 99%