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Cubic graphs admitting transitive non-abelian characteristically simple groups

Part of: Permutation groups Graph theory

Published online by Cambridge University Press:  19 January 2011

Xiao-Hui Hua  and
Yan-Quan Feng
Xiao-Hui Hua
Affiliation:
Department of Mathematics, Beijing Jiaotong University, Beijing 100044, People's Republic of China (yqfeng@bjtu.edu.cn)
Yan-Quan Feng
Affiliation:
Department of Mathematics, Beijing Jiaotong University, Beijing 100044, People's Republic of China (yqfeng@bjtu.edu.cn)
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Abstract

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Let Γ be a graph and let G be a vertex-transitive subgroup of the full automorphism group Aut(Γ) of Γ. The graph Γ is called G-normal if G is normal in Aut(Γ). In particular, a Cayley graph Cay(G, S) on a group G with respect to S is normal if the Cayley graph is R(G)-normal, where R(G) is the right regular representation of G. Let T be a non-abelian simple group and let G = Tℓ with ℓ ≥ 1. We prove that if every connected T-vertex-transitive cubic symmetric graph is T-normal, then every connected G-vertex-transitive cubic symmetric graph is G-normal. This result, among others, implies that a connected cubic symmetric Cayley graph on G is normal except for T ≅ A47 and a connected cubic G-symmetric graph is G-normal except for T ≅ A7, A15 or PSL(4, 2).

Keywords

Cayley graphsymmetric graphquasiprimitive graph

MSC classification

Secondary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 1 , February 2011 , pp. 113 - 123
Copyright
Copyright © Edinburgh Mathematical Society 2011

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