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

Abstract

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 $1\le r\le k\le n$, 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 $1\le r=k\le n$ 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.

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