Integer eigenvalues of the $n$-Queens graph

TL;DR

This paper investigates the integer eigenvalues of the $n$-Queens graph, revealing new eigenvalues and their multiplicities, and providing eigenvectors and conjectures about the spectrum.

Contribution

It proves the existence of new integer eigenvalues for the $n$-Queens graph and analyzes their multiplicities, extending previous spectral results.

Findings

$n-4$ is an eigenvalue with multiplicity at least $(n+1)/2$ or $(n-2)/2$

Additional integer eigenvalues include $-3,-2,\

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Abstract

The $n$-Queens graph, $\mathcal{Q}(n)$, is the graph obtained from a $n\times n$ chessboard where each of its $n^2$ squares is a vertex and two vertices are adjacent if and only if they are in the same row, column or diagonal. In a previous work the authors have shown that, for $n\ge4$, the least eigenvalue of $\mathcal{Q}(n)$ is $-4$ and its multiplicity is $(n-3)^2$. In this paper we prove that $n-4$ is also an eigenvalue of $\mathcal{Q}(n)$ and its multiplicity is at least $\frac{n+1}{2}$ or $\frac{n-2}{2}$ when $n$ is odd or even, respectively. Furthermore, when $n$ is odd, it is proved that $-3,-2\ldots,\frac{n-11}{2}$ and $\frac{n-5}{2},\ldots,n-5$ are additional integer eigenvalues of $\mathcal{Q}(n)$ and a family of eigenvectors associated with them is presented. Finally, conjectures about the multiplicity of the aforementioned eigenvalues and about the non-existence of any other integer eigenvalue are stated.