Arc-transitive Cayley graphs on nonabelian simple groups with prime valency

Yin, Fu-Gang; Yan, Feng; Zhou, Jin‐Xin; Chen, Shanshan

doi:10.1016/j.jcta.2020.105303

Cited by 16 publications

(7 citation statements)

References 17 publications

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“…From the above mentioned results in [6,7,20,29,30,31] on s-arc-transitive nonnormal Cayley graphs on nonabelian simple groups of certain valencies one can observe an interesting phenomenon: most of these graphs turn out to be Cayley graphs on alternating groups. This motivates a natural problem as follows.…”

Section: Introductionmentioning

confidence: 94%

“…Similarly, Du, Feng and Zhou [7] showed that if Val(Γ) = 5 then Γ is an A n+1 -arc-transitive Cayley graph on A n where n is among 11 possible numbers, and [20] showed that if Val(Γ) = 7 and the vertex stabilizer is solvable then Γ is an A n+1 -arc-transitive Cayley graph on A n with n ∈ {7, 21, 63, 84}. Very recently, Yin, Feng, Zhou and Chen [31] proved that if Val(Γ) is a prime greater than 7 and the vertex stabilizer is solvable, then Γ is either an A n+1 -arc-transitive Cayley graph on A n or one of the three exceptions.…”

Section: Introductionmentioning

confidence: 99%

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2-Arc-transitive Cayley graphs on alternating groups

Pan¹,

Xia²,

Yin³

2021

Preprint

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An interesting fact is that most of the known connected 2-arc-transitive nonnormal Cayley graphs of small valency on finite simple groups are (A n+1 , 2)-arc-transitive Cayley graphs on A n . This motivates the study of 2-arc-transitive Cayley graphs on A n for arbitrary valency. In this paper, we characterize the automorphism groups of such graphs. In particular, we show that for a non-complete (G, 2)-arc-transitive Cayley graph on A n with G almost simple, the socle of G is either A n+1 or A n+2 . We also construct the first infinite family of (A n+2 , 2)-arc-transitive Cayley graphs on A n .

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Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

2-Arc-transitive Cayley graphs on alternating groups

Pan¹,

Xia²,

Yin³

2021

Preprint

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“…Normality of Cayley (di)graphs is very important because the automorphism groups of normal Cayley (di)graph A c c e p t e d m a n u s c r i p t are actually known; see Godsil [21] or Proposition 2.2. Furthermore, the study of normality of Cayley (di)graphs is currently a hot topic in algebraic graph theory, and we refer to [14,15,17,18,19,34,45,46] for examples.…”

Section: Introductionmentioning

confidence: 99%

Normal Cayley digraphs of dihedral groups with CI-property

2023

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A Cayley (di)graph Cay(G, S) of a group G with respect to a set S ⊆ G is said to be normal if the image of G under its right regular representation is normal in the automorphism group of Cay(G, S), and is called a CI-(di)graph if for every T ⊆ G with Cay(G, S) ∼ = Cay(G, T ), there is α ∈ Aut(G) such that S α = T . A finite group G is called a DCI-group or an NDCI-group if all Cayley digraphs or all normal Cayley digraphs of G are CI-digraphs, respectively, and is called a CI-group or an NCI-group if all Cayley graphs or all normal Cayley graphs of G are CI-graphs, respectively.Motivated by a conjecture proposed by Ádám in 1967, CI-groups and DCI-groups have been actively studied during the last fifty years by many researchers in algebraic graph theory. It took about thirty years to obtain the classification of cyclic CI-groups and DCIgroups, and recently, the first two authors, among others, classified cyclic NCI-groups and NDCI-groups. Even though there are many partial results on dihedral CI-groups and DCIgroups, their classification is still elusive. In this paper, we prove that a dihedral group of order 2n is an NCI-group or an NDCI-group if and only if n = 2, 4 or n is odd. As a direct consequence, we have that if a dihedral group D 2n of order 2n is a DCI-group then n = 2 or n is odd-square-free, and that if D 2n is a CI-group then n = 2, 9 or n is odd-square-free, throwing some new light on classification of dihedral CI-groups and DCI-groups. As a byproduct, we construct a non-CI Cayley graph of the dihedral group D 8 , but Holt and Royle [A census of small transitive groups and vertex-transitive graphs, J. Symbolic Comput. 101 (2020) 51-60] claimed that D 8 is a CI-group by an algorithm there.

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“…Dobson [9] determined all non-normal Cayley graphs of order a product of two distinct primes, and Dobson and Witte [10] determined all non-normal Cayley graphs of order a prime-square. For normality of Cayley graphs of finite simple groups, we refer the reader to [29,20,16,17,18,45,46,19,13,12,36,47], and for some results on normality of Cayley digraphs, one may see [6,23,21,48]. Based on these results, Xu [44] conjectured that almost all connected Cayley graphs are normal.…”

Section: Introductionmentioning

confidence: 99%

Normal Cayley digraphs of cyclic groups with CI-property

Jin-Hua¹,

Yan²,

Ryabov³

et al. 2021

Preprint

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A Cayley (di)graph Cay(G, S) of a group G with respect to a subset S of G is called normal if the right regular representation of G is a normal subgroup in the full automorphism group of Cay(G, S), and is called a CI-(di)graph if for any Cayley (di)graph Cay(G, T ), Cay(G, S) ∼ = Cay(G, T ) implies that there isgroup or a NDCI-group if all Cayley digraphs or normal Cayley digraphs of G are CI-digraphs, and is called a CI-group or a NCIgroup if all Cayley graphs or normal Cayley graphs of G are CI-graphs, respectively.DCI-groups and CI-groups have been widely studied in the literature. A lot of efforts had been made to attach the classifications of cyclic DCI-groups and CI-groups, motivated by the Ádám's conjecture, and finally Muzychuk finished the classifications by proving that a cyclic group of order n is a DCI-group if and only if n = mk where m = 1, 2, 4 and k is odd square-free, and is a CI-group if and only if either n = 8, 9, 18, or n = mk where m = 1, 2, 4 and k is odd square-free. In this paper we prove that a cyclic group of order n is a NDCI-group if and only if 8 ∤ n, and is a NCI-group if and only if either n = 8 or 8 ∤ n.

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Arc-transitive Cayley graphs on nonabelian simple groups with prime valency

Cited by 16 publications

References 17 publications

2-Arc-transitive Cayley graphs on alternating groups

2-Arc-transitive Cayley graphs on alternating groups

Normal Cayley digraphs of dihedral groups with CI-property

Normal Cayley digraphs of cyclic groups with CI-property

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