Spectral Theory of the Toroidal 3D Queen Graph

TL;DR

This paper analyzes the spectral properties of the toroidal 3D queen graph, providing explicit eigenvalues and multiplicities using Fourier analysis and geometric classification.

Contribution

It offers a detailed spectral characterization of the toroidal 3D queen graph, including explicit formulas for eigenvalues and their multiplicities.

Findings

Eigenvalues are explicitly determined by queen directions orthogonal to frequencies.

Eigenvalue multiplicities are given by explicit polynomials in n.

The spectral analysis combines geometric classification with counting identities.

Abstract

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.