On edge-primitive and 2-arc-transitive graphs

Lu, Zaiping

doi:10.48550/arxiv.1812.10880

Cited by 2 publications

(4 citation statements)

References 18 publications

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“…Since T uv is soluble, the lemma follows. By Lemma 3.4 and [19], we have the following result. Lemma 3.5.…”

Section: Graphs With Soluble Vertex-stabilizersmentioning

confidence: 79%

“…Then what will happen if we assume that Γ is 2-arc-transitive? The second author of this paper showed that AutΓ is almost simple if Γ is 2-arc-transitive, see [19]. This allows us to classify 2-arc-transitive and edge-primitive graphs under certain restrictions.…”

Section: Introductionmentioning

confidence: 92%

“…However, by the Atlas [3] and [1, Table 8.14], X does not have a subgroup of the form of X [1] v .PSL 2 (p), a contradiction. For the first pair, if G = X then G = PGL (2,19) and G {u,v} ∼ = D 40 , and so |G uv | = 20; however, by the Atlas [3], G uv is not contained in any insoluble subgroup other than X, a contradiction. Thus, in this case, G, X and X {u,v} are known as in Table 4.…”

Section: Graphs With Insoluble Vertex-stabilizersmentioning

confidence: 99%

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On edge-primitive graphs with soluble edge-stabilizers

Han¹,

Liao²,

Lu³

2019

Preprint

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“…Since T uv is soluble, the lemma follows. By Lemma 3.4 and [19], we have the following result. Lemma 3.5.…”

Section: Graphs With Soluble Vertex-stabilizersmentioning

confidence: 79%

Section: Introductionmentioning

confidence: 92%

Section: Graphs With Insoluble Vertex-stabilizersmentioning

confidence: 99%

See 1 more Smart Citation

On edge-primitive graphs with soluble edge-stabilizers

Han¹,

Liao²,

Lu³

2019

Preprint

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“…1, where (G, E, A, H) is listed in Table 1. O ′ N Lu [20] Theorem 2. Let Γ be an edge-primitive 3-arc-transitive graph with G = Aut(Γ) such that G is an almost simple classical group and for a vertex v, the vertex-stabiliser G v acts faithfully on the set of neighbours of v. Then Γ = Cos(G, H, HgH) is as in Lemma 1.…”

mentioning

confidence: 99%

On edge-primitive 3-arc-transitive graphs

Giudici¹,

King²

2019

Preprint

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This paper begins the classification of all edge-primitive 3-arc-transitive graphs by classifying all such graphs where the automorphism group is an almost simple group with socle an alternating or sporadic group, and all such graphs where the automorphism group is an almost simple classical group with a vertex-stabiliser acting faithfully on the set of neighbours.Edge-primitive graphs, that is, graphs whose automorphism group acts primitively on the set of edges, were first studied by Weiss in 1973 [32] who classified all the edge-primitive graphs of valency three. The study of edge-primitive graphs was reinvigorated in 2010 by the first author and Li [8] by providing a general structure theorem of such graphs and classifying all edge-primitive graphs whose automorphism group contains PSL 2 (q) as a normal subgroup. This has led to all edge-primitive graphs of valencies 4 [10] and 5 [11] being classified, and all those of prime valency and having a soluble edge-stabiliser [24]. Moreover, all edge-primitive graphs of prime power order [22] or which are Cayley graphs on abelian and dihedral groups [23] have been classified.An s-arc in a graph Γ is an (s + 1)-tuple (v 0 , v 1 , . . . , v s ) of vertices such that v i ∼ v i+1 but v i = v i+2 . We say that Γ is s-arc-transitive if the automorphism group of Γ acts transitively on the set of s-arcs. If all vertices of Γ have valency at least two then an s-arc-transitive graph is also (s − 1)-arc-transitive. The study of s-arc-transitive graphs originated in the seminal work of Tutte [30,31], who showed that a graph of valency three is at most 5-arc-transitive. This was later extended by Weiss [33] who showed that a graph of valency at least three is at most 7-arc-transitive. The vertex-primitive 4-arctransitive graphs were classified by Li [16] and all edge-primitive 4-arc-transitive graphs were classified by Li and Zhang [17]. These classifications were enabled by the classification of all vertex-stabiliser, edge-stabiliser pairs for 4-arc-transitive graphs by Weiss [33]. A key part of the latter classification is that the edge-stabiliser is always soluble. Han and Lu [12] have subsequently classified all edge-primitive graphs for almost simple groups with soluble edge-stabilisers.Edge-primitive 2-arc-transitive graphs have been investigated by Lu [20] who showed that if the graph Γ is not complete bipartite then the automorphism group is almost simple, that is, has a unique minimal normal subgroup T and T is a nonabelian simple group. He further showed that if Γ is 3-arc-transitive then either the graph has valency 7 and G v = A 7 or S 7 , or G v acts unfaithfully on the set Γ(v) of neighbours of v, where G = Aut(Γ). Note that for an edge-primitive graph, the edge-stabiliser G {u,v} is maximal in G. Moreover, the arc-stabiliser G u,v is an index two subgroup of G {u,v} and also contained in two other subgroups, namely the vertex-stabilisers G v and G w . These observations make

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On edge-primitive and 2-arc-transitive graphs

Cited by 2 publications

References 18 publications

On edge-primitive graphs with soluble edge-stabilizers

On edge-primitive graphs with soluble edge-stabilizers

On edge-primitive 3-arc-transitive graphs

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