Normal Cayley graphs of finite groups

Wang, Changqun; Wang, Dianjun; Xu, Mingzhi

doi:10.1007/bf02879042

Cited by 25 publications

(16 citation statements)

References 9 publications

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“…In fact the only groups, for which the complete information about the normality of Cayley graphs is available, are the cyclic groups of prime order [1] and the groups of order twice a prime [4]. Wang et al [19] obtained all disconnected normal Cayley graphs. Let Cay(G, S) be a connected cubic Cayley graph on a non-abelian simple group G. Praeger [17] proved that if N Aut(Cay(G,S)) (R(G)) is transitive on edges then the Cayley graph Cay(G, S) is normal, and Fang et al [5] proved that the vast majority of connected cubic Cayley graphs on non-abelian simple groups are normal.…”

Section: Introductionmentioning

confidence: 99%

Automorphism groups of Cayley graphs on symmetric groups with generating transposition sets

Feng

2006

Journal of Combinatorial Theory, Series B

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Let T be a set of transpositions of the symmetric group S n . The transposition graph Tra(T ) of T is the graph with vertex set {1, 2, . . . , n} and edge set {ij | (i j ) ∈ T }. In this paper it is shown that if n 3, then the automorphism group of the transposition graph Tra(T ) is isomorphic to Aut(S n , T ) = { ∈ Aut(S n ) | T = T } and if T is a minimal generating set of S n then the automorphism group of the Cayley graph Cay(S n , T ) is the semiproduct R(S n ) Aut(S n , T ), where R(S n ) is the right regular representation of S n . As a result, we generalize a theorem of Godsil and Royle [C.D. Godsil, G. Royle, Algebraic Graph Theory, Springer, New York, 2001, p. 53] regarding the automorphism groups of Cayley graphs on S n : if T is a minimal generating set of S n and the automorphism group of the transposition graph Tra(T ) is trivial then the automorphism group of the Cayley graph Cay(S n , T ) is isomorphic to S n .

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Section: Introductionmentioning

confidence: 99%

Automorphism groups of Cayley graphs on symmetric groups with generating transposition sets

Feng

2006

Journal of Combinatorial Theory, Series B

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“…The concept of normal Cayley graphs was first introduced by Xu [23], and since then, it has been extensively studied from different perspectives by many authors. Since Wang et al [21] obtained all disconnected normal Cayley graphs, we only need to consider the connected ones. It can be easily obtained from the elementary permutation group theory that a Cayley graph of prime order is normal if the graph is neither empty nor complete.…”

Section: Introductionmentioning

confidence: 99%

Automorphisms of a Family of Cubic Graphs

Zhou

Ghasemi

2013

Algebra Colloq.

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A Cayley graph Cay(G, S) on a group G with respect to a Cayley subset S is said to be normal if the right regular representation R(G) of G is normal in the full automorphism group of Cay(G, S). For a positive integer n, let Γn be a graph with vertexIn this paper, it is shown that Γn is a Cayley graph and its full automorphism group is isomorphic to Z 3 2 S3 for n = 2, and to Z n 2 D2n for n > 2. Furthermore, we determine all pairs of G and S such that Γn = Cay(G, S) is non-normal for G. Using this, all connected cubic non-normal Cayley graphs of order 8p are constructed explicitly for each prime p.2010 Mathematics Subject Classification: 05C25, 20B25

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“…To study the normality of Cayley graphs it suffices to consider connected normal Cayley graphs, because in [11] all disconnected normal Cayley graphs were determined.…”

Section: Ams Mathematics Subject Classification: 20d60 05b25mentioning

confidence: 99%