An infinite family of biprimitive semisymmetric graphs

Du, Shaofei; Maru?i?, Dragan

doi:10.1002/(sici)1097-0118(199911)32:3<217::aid-jgt1>3.0.co;2-m

Cited by 24 publications

(16 citation statements)

References 10 publications

(9 reference statements)

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“…Recall that the girth of Γ is 4 or 6, and if it is 4, then Γ is isomorphic to K 3,3 or Q 3 (see Remark 1.1). It is easy to see that (7) holds when Γ ∼ = K 3,3 , and we have checked…”

Section: Proof Of Theorem Bmentioning

confidence: 97%

“…If |L| = |R| = 0, then BiCay(H, S) will be written for BiCay(H, ∅, ∅, S). Bi-Cayley graphs have been studied from various aspects [3,11,13,14,16,17,18,20,26,32,34], they have been used by constructions of strongly regular graphs [22,28,29,30] and semisymmetric graphs [7,8,25]. The cubic vertex-transitive abelian bi-Cayley graphs have been classified recently by Feng and Zhou [11] (by a cubic graph we mean a regular graph of valency 3).…”

Section: Introductionmentioning

confidence: 99%

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Arc-transitive cubic abelian bi-Cayley graphs and BCI-graphs

Koike¹,

Kovács²

2016

Filomat

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A finite simple graph is called a bi-Cayley graph over a group H if it has a semiregular automorphism group, isomorphic to H, which has two orbits on the vertex set. Cubic vertex-transitive bi-Cayley graphs over abelian groups have been classified recently by Feng and Zhou (Europ. J. Combin. 36 (2014), 679-693). In this paper we consider the latter class of graphs and select those in the class which are also arc-transitive. Furthermore, such a graph is called 0-type when it is bipartite, and the bipartition classes are equal to the two orbits of the respective semiregular automorphism group. A 0-type graph can be represented as the graph BiCay(H, S), where S is a subset of H, the vertex set of which consists of two copies of H, say H 0 and H 1 , and the edge set is {{h 0 , g 1 } : h, g ∈ H, gh −1 ∈ S}. A bi-Cayley graph BiCay(H, S) is called a BCI-graph if for any bi-Cayley graph BiCay(H, T ), BiCay(H, S) ∼ = BiCay(H, T ) implies that T = hS α for some h ∈ H and α ∈ Aut(H). It is also shown that every cubic connected arc-transitive 0-type bi-Cayley graph over an abelian group is a BCI-graph.

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“…Recall that the girth of Γ is 4 or 6, and if it is 4, then Γ is isomorphic to K 3,3 or Q 3 (see Remark 1.1). It is easy to see that (7) holds when Γ ∼ = K 3,3 , and we have checked…”

Section: Proof Of Theorem Bmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Arc-transitive cubic abelian bi-Cayley graphs and BCI-graphs

Koike¹,

Kovács²

2016

Filomat

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“…We are grateful to several colleagues who kindly provided us with requested references and suggested new ones: to M. H. Klin for [10], [18], and [19], to D. Marušič for [14], and to Ming-Yao Xu for [4] and [6]. We would also like to thank D. Marušič and C. Wang for their helpful correspondence.…”

Section: Acknowledgmentsmentioning

confidence: 99%

“…The list of open questions at the end of this paper stimulated the related research done by others in the many years to come. Most of these questions have been answered, but new and related questions were asked and the investigation continues (see [1][2][3][4][5][6], [8][9][10], [14][15][16][17][18][19]). Though there are some general group-theoretic approaches for construction of semisymmetric graphs, explicit examples of infinite series are rare.…”

Section: Introductionmentioning

confidence: 99%

An infinite series of regular edge‐ but not vertex‐transitive graphs

Lazebnik

Viglione

2002

Journal of Graph Theory

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“…However, graphs with various symmetries can be constructed by bi-Cayley graphs. The smallest trivalent semisymmetric graph is the Gray graph [6], which is a bi-Cayley graph over a non-abelian metacyclic group of order 27, and infinite semisymmetric graphs were constructed in [17,18,37]. Boben et al [5] studied properties of cubic bi-Cayley graphs over cyclic groups and the configurations arising from these graphs.…”

Section: Introductionmentioning

confidence: 99%

Bipartite edge-transitive bi-p-metacirculants

Yan

Wang

2019

Ars Math. Contemp.

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A graph is a bi-Cayley graph over a group if the group acts semiregularly on the vertex set of the graph with two orbits. Let G be a non-abelian metacyclic p-group for an odd prime p. In this paper, we prove that if G is a Sylow p-subgroup in the full automorphism group Aut(Γ) of a graph Γ, then G is normal in Aut(Γ). As an application, we classify the half-arc-transitive bipartite bi-Cayley graphs over G of valency less than 2p, while the case for valency 4 was given by Zhang and Zhou in 2019. It is further shown that there are no semisymmetric or arc-transitive bipartite bi-Cayley graphs over G of valency less than p.

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An infinite family of biprimitive semisymmetric graphs

Cited by 24 publications

References 10 publications

Arc-transitive cubic abelian bi-Cayley graphs and BCI-graphs

Arc-transitive cubic abelian bi-Cayley graphs and BCI-graphs

An infinite series of regular edge‐ but not vertex‐transitive graphs

Bipartite edge-transitive bi-p-metacirculants

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