LP relaxations and Fuglede's conjecture

Consider a unitary (up to scaling) submatrix of the Fourier matrix with rows indexed by $\mathcal{I}$ and columns indexed by $\mathcal{J}$. From the column index set $\mathcal{J}$ we construct a graph $\mathcal{G}$ so that the row index set $\mathcal{I}$ determines a max-clique. Interpreting $\mathcal{G}$ as coming from an association scheme gives certain bounds on the clique number, which has possible applications to Fuglede's conjecture on spectral and tiling sets.