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On 2-arc-transitivity of Cayley graphs

2003
*
Journal of combinatorial theory. Series B (Print)
*

The classification

doi:10.1016/s0095-8956(02)00033-3
fatcat:vqaezpf3pffmzjfmph7oj3wfm4
*of**2*-*arc*-*transitive**Cayley**graphs**of*cyclic groups, given in (J. Algebra. ... Combin. 5 (1996) 83-86) by Alspach, Conder, Xu and the author, motivates the main theme*of*this article: the study*of**2*-*arc*-*transitive**Cayley**graphs**of*dihedral groups. ... In Section*2*we start by giving a brief history*of*the research*on**arc*-*transitive**graphs*and by laying out the basic strategy behind the analysis*of**2*-*arc*-*transitive**Cayley**graphs**of*dihedral groups. ...##
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On 2-arc-transitive Cayley graphs of Abelian groups

2002
*
Discrete Mathematics
*

A

doi:10.1016/s0012-365x(01)00068-1
fatcat:nld4io737zfvnpkm5jjzd2dhqi
*graph*X is*2*-*arc*-*transitive*if its automorphism group acts*transitively**on*the set*of**2*-*arcs**of*X . Some properties*of**2*-*arc*-*transitive**Cayley**graphs**of*Abelian groups are considered. ... It is also proved that the set*of*generators*of*a*2*-*arc*-*transitive**Cayley**graph**of*an Abelian group which is not a circulant contains no elements*of*odd order. ... Some other examples*of**2*-*arc*-*transitive**Cayley**graphs**of*Abelian groups were given in [*2*] where certain subclass*of**Cayley**graphs**of*elementary Abelian groups were considered. ...##
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A Class of Arc-Transitive Cayley Graphs as Models for Interconnection Networks

2009
*
SIAM Journal on Discrete Mathematics
*

We study a class

doi:10.1137/06067434x
fatcat:5jxab4srovck5jtssetfanqb64
*of**Cayley**graphs*as models for interconnection networks. ... In particular, these hold for two important subclasses*of**graphs*, namely,*Cayley**graphs*admitting a complete rotation and Frobenius*graphs**of*a certain type. ... Figure*2*:*2*Classes*of**Cayley**graphs*considered and their relations. ...##
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Arc-Transitive Cayley Graphs of Valency at Most Four on Abelian Groups

2001
*
Southeast Asian Bulletin of Mathematics
*

In this paper we give a complete classi®cation for

doi:10.1007/s10012-001-0355-z
fatcat:ljpxbf5rxzfbjnogsiiksrkjqm
*arc*-*transitive*and*one*-regular*Cayley**graphs**of*valency at most four*on*®nite abelian groups. ... Then all*arc*-*transitive**Cayley**graphs**of*valency 3 and 4*on*G are listed in Propositions 3.1 and 3.3.All*one*-regular*Cayley**graphs**of*valency 4*on*G are listed in Proposition 3.3(*2*)(iv).The next theorem ... Let H hsiÂhti, then H q Z*2**2*. r The next proposition determines all normal,*arc*-*transitive**Cayley**graphs**of*valency 3 and 4*on*a non-cyclic abelian groups. ...##
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Normal edge-transitive and \frac{1}{2}-arc-transitive cayley graphs on non-abelian groups of odd order 3pq, p and q are primes

2018
*
Tamkang Journal of Mathematics
*

In this paper, the connected

doi:10.5556/j.tkjm.49.2018.2169
fatcat:qi27ogdtefbqlde3ojdbdqfhqa
*Cayley**graph**of*groups*of*order 3p q, for primes p and q, are investigated and all connected normal 1*2*−*arc*-*transitive**Cayley**graphs**of*group*of*these orders will be classified ... The research*of*the authors are partially supported by the University*of*Kashan under grant no 159020/303. ... −*arc**transitive**Cayley**graph*. ...##
###
Corrigendum to "On 2-arc-transitivity of Cayley graphs" [J. Combin. Theory Ser. B 87 (2003) 162–196]

2006
*
Journal of combinatorial theory. Series B (Print)
*

A characterization

doi:10.1016/j.jctb.2006.01.003
fatcat:xc3pzitzlra7xgo4iapbz3q3na
*of**2*-*arc*-*transitive*dihedrants, that is,*Cayley**graphs**of*dihedral groups, is given in [3, Theorem 8.4], describing such*graphs*either explicitly or as cyclic covers*of*certain basic ... Consequently, the automorphism group A*of*a connected*2*-*arc*-*transitive**Cayley**graph*X K 2n*of*a dihedral group D 2n = ρ, τ | ρ n = τ*2*= (ρτ )*2*= 1 is necessarily imprimitive, allowing the above mentioned ... Let n 3, and let X be a connected,*2*-*arc*-*transitive**Cayley**graph**of*a dihedral group D = D 2n = ρ, τ | ρ n = τ*2*= (ρτ )*2*= 1*of*order 2n. ...##
###
On finite dual Cayley graphs

2020
*
Open Mathematics
*

A

doi:10.1515/math-2020-0141
fatcat:auxp6aornzd4lk36hlrjawmcwq
*Cayley**graph*\Gamma*on*a group G is called a dual*Cayley**graph**on*G if the left regular representation*of*G is a subgroup*of*the automorphism group*of*\Gamma (note that the right regular representation ... In this article, we study finite dual*Cayley**graphs*regarding identification, construction,*transitivity*and such*graphs*with automorphism groups as small as possible. ... Funding: This work was partially supported by National Natural Science Foundation*of*China (11961076 and 11461007). ...##
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Normal Edge-Transitive Cayley Graphs of the Group U6n

2014
*
International Journal of Combinatorics
*

A

doi:10.1155/2014/628214
fatcat:cys35jaiqrhufafoqhnxouaf6i
*Cayley**graph**of*a group G is called normal edge-*transitive*if the normalizer*of*the right representation*of*the group in the automorphism*of*the*Cayley**graph*acts*transitively**on*the set*of*edges*of*... In this paper, we determine all connected normal edge-*transitive**Cayley**graphs**of*the group U6n. ... Aut(Γ) ( ( )) acts*on*the edges or*arcs**of*Γ, respectively, and it is called normal halfarc-*transitive**Cayley**graph*if it is normal edge-*transitive**Cayley**graph*which is not normal*arc*-*transitive*. ...##
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Arc-transitive pentavalent Cayley graphs with soluble vertex stabilizer on finite nonabelian simple groups
[article]

2017
*
arXiv
*
pre-print

To our knowledge, this is the first known example

arXiv:1702.05754v1
fatcat:ju7gzugukbe67ne4ju6e2anbmi
*of*pentavalent 3-*arc*-*transitive**Cayley**graph**on*finite nonabelian simple group which is non-normal. ... In this paper, we investigate the normality problem*of*the connected*arc*-*transitive*pentavalent*Cayley**graphs*with soluble vertex stabilizer*on*finite nonabelian simple groups. ... This completes the proof*of*the lemma. ...##
###
Cubic s-arc transitive Cayley graphs

2009
*
Discrete Mathematics
*

*graph*

*of*G/ ( 3 ) 3 If N has at leat three orbits

*on*V Γ, then Γ N is an X/N -edge-

*transitive*

*Cayley*

*graph*

*of*G/N and either (a) val(Γ N ) < val(Γ ) and Γ is not (X,

*2*)-

*arc*-

*transitive*; or . ... Then N has at least three orbits

*on*V Γ and Γ is a cover

*of*(the 'basic' (X/N, s)-

*arc*-

*transitive*

*Cayley*

*graph*) Γ N provided s ≥ 3. ... Construction

*of*basic

*arc*-

*transitive*

*Cayley*

*graph*with given vertex stabilizer H and

*arc*stabilizer P < H. Step1: Determine I = {τ ∈ N S n (P ) \ 1 =K¢H N S n (K)|τ

*2*= 1, 1 τ = 1}. ...

##
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Vertex-transitive graphs that have no Hamilton decomposition
[article]

2014
*
arXiv
*
pre-print

It is shown that there are infinitely many connected vertex-

arXiv:1408.5211v3
fatcat:wrijoayk4bfwln472av3au73bq
*transitive**graphs*that have no Hamilton decomposition, including infinitely many*Cayley**graphs**of*valency 6, and including*Cayley**graphs**of*arbitrarily ... The authors acknowledge the support*of*the Australian Research Council via grants DP120100790 and DP120103067, and are grateful to Marston Conder for helpful discussions. ...*of*a*graph*X and the vertices*of*K(L(X)), that Alt(5) has a regular action*on*the*2*-*arcs**of*F 10, and that PSL(*2*, 7) has a regular action*on*the*2*-*arcs**of*F 28, see [14] . ...##
###
On edge-transitive Cayley graphs of valency four

2004
*
European journal of combinatorics (Print)
*

Various new half-

doi:10.1016/j.ejc.2003.07.008
fatcat:4imho3xnwjdnhdhpm4i7n3yg4y
*arc**transitive**graphs*are constructed. A n−1 where n =*2*r 3 s−1 for r ∈ {*2*, 3, 4} ... A characterization is given*of*a class*of*edge-*transitive**Cayley**graphs*, providing methods for constructing*Cayley**graphs*with certain symmetry properties. ... The third author is supported by Natural Science Foundation*of*China and the Research Fund for the Doctoral Program*of*Higher Education. ...##
###
Arc-transitive and s-regular Cayley graphs of valency 5 on Abelian groups

2006
*
Discussiones Mathematicae Graph Theory
*

A

doi:10.7151/dmgt.1328
fatcat:bqfooecg5vgqdjcac2bnzkemfe
*graph*Γ is said to be*arc*-*transitive*, if Aut(Γ) is*transitive**on*an*arc*set. Also, a*graph*Γ is s-regular if Aut(Γ) acts regularly*on*the set*of*s-*arcs*. ... In this paper, we first give a complete classification for*arc*-*transitive**Cayley**graphs**of*valency five*on*finite Abelian groups. ... The purpose*of*this paper is to investigate*arc*-*transitive**Cayley**graphs**of*valency five*on*an Abelian group, that is, the*arc*-*transitive**graphs*whose automorphism groups have an Abelian regular subgroup ...##
###
Finite 2-arc-transitive abelian Cayley graphs

2008
*
European journal of combinatorics (Print)
*

Let Γ be a finite

doi:10.1016/j.ejc.2006.12.001
fatcat:whbecaouvfdp5ocbmp4s6jgv3m
*2*-*arc*-*transitive**Cayley**graph**of*an abelian group. ... In particular,*one**of*three cases occurs: Γ = K n,n − nK*2*where n is even but is not a*2*-power, Γ has*2*-power number*of*vertices, or Γ is a circulant. ... Acknowledgements This work was partially supported by an*ARC*Discovery Grant Project. The author is very grateful to the referees for their constructive comments. ...##
###
Cubic core-free symmetric m-Cayley graphs

2018
*
Journal of Algebraic Combinatorics
*

This generalises the notions

doi:10.1007/s10801-018-0847-x
fatcat:ffo63to5hzgqjle6nluv2v457u
*of*a*Cayley**graph*(where m = 1) and a bi-*Cayley**graph*(where m =*2*). ... We also prove that there is no 3-*arc*-*transitive*normal*Cayley**graph*or bi-*Cayley**graph*(with valency 3 or more), which answers a question posed by Li (Proc Amer Math Soc 133:31-41 2005). ...*Cayley**graphs*admit a group*of*automorphisms that acts regularly (sharply*transitively*)*on*vertices. ...
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