On the minimum supports of some eigenfunctions in the Doob graphs

We prove that the minimum size of the support of an eigenfunction in the Doob graph D(m, n) corresponding to the second largest eigenvalue is 6 · 4^2m+n-2, and obtain characterisation of all eigenfunctions with minimum support. Similar results, with the minimum support size 2^2m+n, are obtained for the minimum eigenvalue of D(m, n).