Spectral Theory of the Toroidal 3D Queen Graph
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911567799386112 |
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| author | Ramani, Mahesh |
| author_facet | Ramani, Mahesh |
| contents | 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 $λ(a)=nμ(a)-13$, where $μ(a)$ 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 $μ(a)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_03842 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Spectral Theory of the Toroidal 3D Queen Graph Ramani, Mahesh Combinatorics Spectral Theory 05C50, 05C25 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 $λ(a)=nμ(a)-13$, where $μ(a)$ 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 $μ(a)$ 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. |
| title | Spectral Theory of the Toroidal 3D Queen Graph |
| topic | Combinatorics Spectral Theory 05C50, 05C25 |
| url | https://arxiv.org/abs/2604.03842 |