Automorphism group of the graph A (n, k, r)

Let n^ (k) be the set of all ordered k-tuples of distinct elements in n=\1, 2,. . . , n\. The (n, k, r) -arrangement graph A (n, k, r) with 1 r k n, is the graph with vertex set n^ (k) and with two k-tuples are adjacent if they differ in exactly r coordinates. In this manuscript, we characterize the full automorphism groups of A (n, k, r) in the cases that 1 r=k n and r=2<k=n. Thus, we resolve two special cases of an open problem proposed by Fu-Gang Yin, Yan-Quan Feng, Jin-Xin Zhou and Yu-Hong Guo. In addition, we conclude with a bold conjecture.