For odd n not divisible by 3 the toroidal 3D queen graph has eigenvalues nμ-13 where μ takes values in {0,1,2,3,4,13} with explicit polynomial multiplicities in n.
Terras,Fourier Analysis on Finite Groups and Applications, London Mathematical Society Student Texts43, Cambridge University Press
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Spectral Theory of the Toroidal 3D Queen Graph
For odd n not divisible by 3 the toroidal 3D queen graph has eigenvalues nμ-13 where μ takes values in {0,1,2,3,4,13} with explicit polynomial multiplicities in n.