Abstract
A Cayley graph \({\varGamma }=\mathsf{Cay}(G,S)\) is said to be normal if G is normal in \(\mathsf{Aut}{\varGamma }\). The concept of normal Cayley graphs was first proposed by Xu (Discrete Math 182:309–319, 1998) and it plays an important role in determining the full automorphism groups of Cayley graphs. In this paper, we study the normality of connected arc-transitive pentavalent Cayley graphs \({\varGamma }\) on finite nonabelian simple groups G, where the vertex stabilizer \(\mathsf{A}_v\) is soluble for \(\mathsf{A}=\mathsf{Aut}{\varGamma }\) and \(v\in V{\varGamma }\). We prove that \({\varGamma }\) is either normal or \(G=\mathsf{A}_{39}\) or \(\mathsf{A}_{79}\). Further, a connected pentavalent arc-transitive non-normal Cayley graph on \(\mathsf{A}_{79}\) is constructed. To our knowledge, this is the first known example of pentavalent 3-arc-transitive Cayley graph on finite nonabelian simple group which is non-normal.
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The authors are very grateful to the referees for their valuable comments and suggestions.
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Bo Ling was supported by the National Natural Science Foundation of China (11701503, 11761079, 11361006), and the Natural Science Foundation of Yunnan Province (2017ZZX086, 2015J006). Ben Gong Lou was supported by the National Natural Science Foundation of China (11231008, 11461004).
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Ling, B., Lou, B.G. On Arc-Transitive Pentavalent Cayley Graphs on Finite Nonabelian Simple Groups. Graphs and Combinatorics 33, 1297–1306 (2017). https://doi.org/10.1007/s00373-017-1845-9
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DOI: https://doi.org/10.1007/s00373-017-1845-9