Spectral Theory of the Toroidal 3D Queen Graph

We study the adjacency spectrum of the toroidal three-dimensional queen graph $G_n$ on $({\mathbb{Z}}_n{)}^3$. Since $G_n$ is a Cayley graph on an abelian group, its adjacency matrix is diagonalized by Fourier characters. For each frequency $a\in ({\mathbb{Z}}_n{)}^3$, the corresponding eigenvalue is $\lambda (a)=n\mu (a)-13$, where $\mu (a)$ counts the queen directions orthogonal to $a$ modulo $n$. In the generic odd case, meaning $n$ odd with $3\nmid n$, the possible values of $\mu (a)$ are exactly $0,1,2,3,4,$ and $13$, and each multiplicity is given by an explicit polynomial in $n$. The proof combines a geometric classification of frequency points by orthogonality type with two global counting identities.