Automorphisms and isomorphisms of enhanced hypercubes

LetZ2 be the elementary abelian 2-group, which can be viewed as the vector space of dimension n over F2. Let {e1, . . . , en} be the standard basis of Z2 and k = ek + · · · + en for some 1 ≤ k ≤ n − 1. Denote by Γn,k the Cayley graph over Z2 with generating set Sk = {e1, . . . , en, k}, that is, Γn,k = Cay(Z2 ,Sk). In this paper, we characterize the automorphism group of Γn,k for 1 ≤ k ≤ n − 1 and determine all Cayley graphs over Z2 isomorphic to Γn,k. Furthermore, we prove that for any Cayley graph Γ = Cay(Z n 2 ,T), if Γ and Γn,k share the same spectrum, then Γ Γn,k. Note that Γn,1 is known as the so called n-dimensional folded hypercube FQn, and Γn,k is known as the n-dimensional enhanced hypercube Qn,k.