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

Abstract

The ( n , k )‐arrangement graph A ( n , k ) with 1 ≤ k ≤ n − 1, is the graph with vertex set the ordered k‐tuples of distinct elements in { 1 , 2 , … , n } and with two k‐tuples adjacent if they differ in exactly one of their coordinates. The ( n , k )‐arrangement graph was proposed by Day and Tripathi in 1992, and is a widely studied interconnection network topology. The Johnson graph J ( n , k ) with 1 ≤ k ≤ n − 1, is the graph with vertex set the k‐element subsets of { 1 , 2 , … , n }, and with two k‐element subsets adjacent if their intersection has k − 1 elements. In 1989, Brouwer, Cohen and Neumaier determined the automorphism group of J ( n , k ), and in 2015, Dobson and Malnič proved that J ( n , k ) a is Cayley graph if and only if ( n , k ) = ( 8 , 3 ), ( 32 , 3 ) or ( n , 2 ) with n ≡ 3 ( sans-serifmod 4 ) being a prime‐power. In this article we prove that sans-serifAut ( A ( n , k ) ) ≅ S n × S k, and as a byproduct, A ( n , k ) is a normal cover of J ( n , k ). Furthermore, A ( n , k ) is a Cayley graph if and only if ( n , k ) = ( 33 , 4 ), ( 11 , 4 ), ( 9 , 4 ), ( 12 , 5 ), ( 8 , 5 ), ( 9 , 6 ), ( 32 , 29 ), ( 33 , 30 ), ( n , 1 ), ( n , n − 1 ), ( n , n − 2 ), ( q , 2 ) or ( q + 1 , 3 ), where q is a prime‐power. Note that the graph A ( n , n − 1 ) is called the n‐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 A ( n , n − 2 ) is a Cayley graph on the alternating group A n, and in 2011, Zhou determined the automorphism group of A ( n , n − 2 ).