The isomorphism problem for rose window graphs

Given integers n ≥ 3 and 1 ≤ a , r ≤ n − 1 with r ≠ n / 2 , the rose window graph R n ( a , r ) is the graph with vertex set { A i , B i | i ∈ { 0 , 1 , … , n − 1 } } and edges { A i , A i + 1 } , { A i , B i } , { A i + a , B i } and { B i , B i + r } for every i ∈ { 0 , 1 , … , n − 1 } where addition in subscripts is modulo n . In this paper we give necessary and sufficient conditions for two rose window graphs to be isomorphic.