  Let $A_n$ be the alternating group of even permutations of $X:=\{1,2,...,n\}$
and ${\mathcal E}_n$ the set of even derangements on $X.$ Denote by $A\T_n^q$
the tensor product of $q$ copies of $A\T_n,$ where the Cayley graph
$A\T_n:=\T(A_n,{\mathcal E}_n)$ is called the even derangement graph. In this
paper, we intensively investigate the properties of $A\T_n^q$ including
connectedness, diameter, independence number, clique number, chromatic number
and the maximum-size independent sets of $A\T_n^q.$ By using the result on the
maximum-size independent sets $A\T_n^q$, we completely determine the full
automorphism groups of $A\T_n^q.$
