The symmetry property of (n,k)‐arrangement graph

Abstract The ‐ arrangement graph with , is the graph with vertex set the ordered ‐tuples of distinct elements in and with two ‐tuples adjacent if they differ in exactly one of their coordinates. The ‐arrangement graph was proposed by Day and Tripathi in 1992, and is a widely studied interconnection network topology. The Johnson graph with , is the graph with vertex set the ‐element subsets of , and with two ‐element subsets adjacent if their intersection has elements. In 1989, Brouwer, Cohen and Neumaier determined the automorphism group of , and in 2015, Dobson and Malnič proved that a is Cayley graph if and only if , or with being a prime‐power. In this article we prove that , and as a byproduct, is a normal cover of . Furthermore, is a Cayley graph if and only if , , , , , , , , , , , or , where is a prime‐power. Note that the graph is called the ‐ star graph , and its automorphism group can be deduced from a general result given by Feng in 2006. In 1998, Chiang and Chen proved that is a Cayley graph on the alternating group , and in 2011, Zhou determined the automorphism group of .