Edge-primitive Cayley graphs on abelian groups and dihedral groups

Pan, Jiangmin; Wu, Ci Xuan; Yin, Fu-Gang

doi:10.1016/j.disc.2018.08.023

Cited by 7 publications

(7 citation statements)

References 18 publications

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“…Guo et al classified edge-primitive tetravalent and pentavalent graphs in [5] and [6]. Pan et al discussed edge-primitive graphs of prime valency in [7], and edge-primitive Cayley graphs on abelian groups and dihedral groups in [8]. Lu [9] proved that a finite 2-arc-transitive edge-primitive graph has an almost simple automorphism group if it is neither a cycle nor a complete bipartite graph.…”

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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“…Research on Cayley graphs is an intriguing topic for graph algebra theory research and study. Here are various studies on algebraic graphs, such as the primitive edge of the Cayley graph on the abelian group and the dihedral group, which describes an edge-primitive graph if the automorphism group operates primitively on the edge set [5]. Cayley graph illustrating the symmetry group and the number of isomorphic graph patterns produced [6].…”

Section: Introductionmentioning

confidence: 99%

“…Hamilton decomposition of graph Cay(D 2n , H ) , sr , sr 3 , sr 2 sr 3 , r 2 , r 3 , r 2 , r 4 , r 3 , , sr , sr 3 , sr 2 sr 4 , sr3 , sr 4 , r 2 , r 3 , r 2 , r 4 , r 3 , r 5 , r 4 , 1, r 5 , r , r 3 , sr 3 , r 4 , sr 2 , r 5 , sr , sr5 , sr4 …”

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confidence: 99%

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Decomposition Hamilton in Cayley Graphs with Certain Invers Generator of Dihedral-$$2n$$ Group

Kumala,

Susanto,

Rahmadani

et al. 2023

Advances in Engineering Research

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The Hamilton decomposition of graph G is a partition of the edge set into a Hamilton cycle and 1-factor if the vertex degree is odd or a partition into a Hamilton cycle if the degree of the vertex is even. In 2020, the focus was on determining the Hamilton decomposition of a Cayley graph in the dihedral-2p group, where p is a single prime. In this paper, the Hamiltonian decomposition of the Cayley graph will be determined from the dihedral-2n group, with n ≥ 3. This research aims to determine Hamiltonian decomposition, which focuses on generating {r n−1 , s} from dihedral groups. The research method is to determine the dihedral-2n group, determine the set of vertices and the set of edges of the Cayley graph, construct the Cayley graph generated by {r n−1 , s}, and decompose the Cayley graph using Hamiltonian. The vertex degree of the Cayley graph, constructed by the dihedral-2n group and generated by {r n−1 , s}, , is an odd vertex. Graph decomposition results are a Hamilton cycle and 1-factor (perfect matching).

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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 , . .…”

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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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Edge-primitive Cayley graphs on abelian groups and dihedral groups

Cited by 7 publications

References 18 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

Decomposition Hamilton in Cayley Graphs with Certain Invers Generator of Dihedral-$$2n$$ Group

On edge-primitive 3-arc-transitive graphs

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