We study the problem of finding unitary submatrices of the N × N discrete Fourier transform matrix, in the context of interpolating a discrete bandlimited signal using an orthogonal basis. This problem is related to a diverse set of questions on idempotents on ZN and tiling ZN. In this work, we establish a graph-theoretic approach and connections to the problem of finding maximum cliques. We identify the key properties of these graphs that make the interpolation problem tractable when N is a prime power, and we identify the challenges in generalizing to arbitrary N. Finally, we investigate some connections between graph properties and the spectral-tile direction of the Fuglede conjecture. © 1963-2012 IEEE.