Cospectral mates for the union of some classes in the Johnson association scheme (1701.08747v2)

Abstract: Let $n\geq k\geq 2$ 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 $|A\cap B|\in S$. In this paper, we use Godsil-McKay switching to prove that for $m\geq 0$, $k\geq \max(m+2,3)$ and $S = {0, 1, ..., m}$, the graphs $J_S(3k-2m-1,k)$ are not determined by spectrum and for $m\geq 2$, $n\geq 4m+2$ and $S = {0,1,...,m}$ the graphs $J_{S}(n,2m+1)$ 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 $k\leq 5$.