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A Cayley graph $${\varGamma }=\mathsf{Cay}(G,S)$$ Γ = Cay ( G , S ) is said to be normal if G is normal in $$\mathsf{Aut}{\varGamma }$$ Aut Γ . 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...
Hua et al. (Discrete Math 311, 2259–2267, 2011) and Yang et al. (Discrete Math. 339, 522–532, 2016) classify arc-transitive pentavalent graphs of order 2pq and of order 2pqr (with p, q, r distinct odd primes), respectively. In this paper, we extend their results by giving a classification of arc-transitive pentavalent graphs of any square-free order.
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