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Spectral Theory of the Toroidal 3D Queen Graph

Mahesh Ramani

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 $λ(a)=nμ(a)-13$, 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.

Spectral Theory of the Toroidal 3D Queen Graph

Abstract

We study the adjacency spectrum of the toroidal three-dimensional queen graph on . Since is a Cayley graph on an abelian group, its adjacency matrix is diagonalized by Fourier characters. For each frequency , the corresponding eigenvalue is , where counts the queen directions orthogonal to modulo . In the generic odd case, meaning odd with , the possible values of are exactly and , and each multiplicity is given by an explicit polynomial in . The proof combines a geometric classification of frequency points by orthogonality type with two global counting identities.

Paper Structure

This paper contains 6 sections, 5 theorems, 27 equations, 1 table.

Table of Contents

  1. Introduction
  2. The toroidal 3D queen graph
  3. Fourier diagonalization and the eigenvalue formula
  4. Orbit types and the structure of multiplicities
  5. Example
  6. Concluding remarks

Key Result

Theorem 3.1

Assume $n$ is odd and $3\nmid n$. For every $a\in(\mathbb{Z}_n)^3$, the character $\chi_a$ is an eigenvector of the adjacency matrix $A$ of $G_n$ with eigenvalue $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (12)

  • Definition 2.1
  • Theorem 3.1: Eigenvalue formula
  • proof
  • Corollary 3.2
  • proof
  • Corollary 3.3
  • proof
  • Lemma 4.1: Prototype lines, explicit orbit check
  • proof
  • Theorem 4.2: Exact multiplicities in the generic odd case
  • ...and 2 more