Abstract
In this paper, we prove that the Cayley digraph Γ = Cay(G, S) of valency 2 on non-abelian group G of odd order is normal if the automorphism group of A(Γ), a graph constructed from Γ by using the method presented in the paper, is primitive on the vertices set V(A(Γ). We also prove that the Cayley digraphs of valency 2 on non-abelian group of order pq2 are normal, where p and q are distinct odd primes.
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