  We study the problem of finding unitary submatrices of the $N \times N$
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 $\mathbb{Z}_N$ and tiling
$\mathbb{Z}_N$. 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.
