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 $6 \cdot 4^{2m+n-2}$, 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 $D(m,n)$.