Minimum supports of eigenfunctions of Johnson graphs

We study the weights of eigenvectors of the Johnson graphs $J(n,w)$. For any $i\in \{1,\dots ,w\}$ and sufficiently large $n,n\ge n(i,w)$ we show that an eigenvector of $J(n,w)$ with the eigenvalue ${\lambda }_i=(n-w-i)(w-i)-i$ has at least $2^i{(}_{w-i}^{n-2i})$ nonzeros and obtain a characterization of eigenvectors that attain the bound.