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

Let $n\ge k\ge 2$ be two integers and $S$ a subset of $\{0,1,\dots ,k-1\}$. The graph $J_S(n,k)$ has as vertices the $k$-subsets of the $n$-set $[n]=\{1,\dots ,n\}$ and two $k$-subsets $A$ and $B$ are adjacent if $|A\cap B|\in S$\mathrm{.}Inthispaper,weuseGodsil-McKayswitchingtoprovethatfor$m\geq 0$,$k\geq \max(m+2,3)$and$S = {0, 1, …, m}$,thegraphs$J_S(3k-2m-1,k)$arenotdeterminedbyspectrumandfor$m\geq 2$,$n\geq 4m+2$and$S = {0,1,…,m}$thegraphs$J_{S}(n,2m+1)$ 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 $k\le 5$.