Edge-Primitive Graphs of Prime Power Order

Pan, Jiangmin; Huang, Zhaohong; Wu, Ci Xuan

doi:10.1007/s00373-018-1997-2

Cited by 7 publications

(2 citation statements)

References 17 publications

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“…The work of studying edge-primitive graphs of specific orders is also attractive. Pan et al studied all edge-primitive graphs of prime power order in [11], and edge-primitive graphs of order twice as a prime power in [12]. The main work of this paper is to classify all edge-primitive graphs of order as a product of two distinct primes.…”

Section: Introductionmentioning

confidence: 99%

On Edge-Primitive Graphs of Order as a Product of Two Distinct Primes

Xiao,

Zhang,

Zhang

2023

Mathematics

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Section: Introductionmentioning

confidence: 99%

On Edge-Primitive Graphs of Order as a Product of Two Distinct Primes

Xiao,

Zhang,

Zhang

2023

Mathematics

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“…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 , . .…”

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 graphs of order twice a prime power

Pan

2019

Discrete Mathematics

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Edge-Primitive Graphs of Prime Power Order

Cited by 7 publications

References 17 publications

On Edge-Primitive Graphs of Order as a Product of Two Distinct Primes

On Edge-Primitive Graphs of Order as a Product of Two Distinct Primes

On edge-primitive 3-arc-transitive graphs

On edge-primitive graphs of order twice a prime power

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