Abstract
A Cayley graph X = Cay(G, S) on a group G is said to be normal if the right regular representation R(G) of G is normal in the full automorphism group Aut(X). In this paper, two novel conditions are outlined to identify the non-normal Cayley graphs of valency six. As an application, some non-normal Cayley graphs of valency six on A6 and A5 are obtained.
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