Cospectral mates for the union of some classes in the Johnson association scheme

Let n≥k≥2 be two integers and S a subset of {0,1,…,k−1}. The graph JS(n,k) has as vertices the k-subsets of the n-set [n]={1,…,n} and two k-subsets A and B are adjacent if |A∩B|∈S. In this paper, we use Godsil–McKay switching to prove that for m≥0, k≥max⁡(m+2,3) and S={0,1,…,m}, the graphs JS(3k−2m−1,k) are not determined by spectrum and for m≥2, n≥4m+2 and S={0,1,…,m} the graphs JS(n,2m+1) are not determined by spectrum. We also report some computational searches for Godsil–McKay switching sets in the union of classes in the Johnson scheme for k≤5.