Spectral properties of the n-Queens' graphs

Abstract

The n-Queens’ graph, Q(n), is the graph associated to the n×n chessboard (a generalization of the classical 8×8 chessboard), with n 2 vertices, each one corresponding to a square of the chessboard. Two vertices of Q(n) are adjacent if and only if they are in the same row, in the same column or in the same diagonal of the chessboard. After a short overview on the main combinatorial properties of Q(n), its spectral properties are investigated. First, a lower bound on the least eigenvalue of an arbitrary graph is obtained using clique edge partitions and a sufficient condition for this lower bound be attained is deduced. For the particular case of Q(n), we prove that for every n, its least eigenvalue is not less than −4 and it is equal to −4 with multiplicity (n − 3)2 , for every n ≥ 4. Furthermore, n − 4 is also an eigenvalue of Q(n), with multiplicity at least n−2 2 when n is even and at least n+1 2 when n is odd. A conjecture about the integer eigenvalues of Q(n) is presented. We finish this article with an algorithm to determine an equitable partition of the n-Queens’ graph, Q(n), for n ≥ 3, concluding that such equitable partition has (⌈n/2⌉+1)⌈n/2⌉ 2 cells.

The $n$-Queens' graph, $\mathcal{Q}(n)$, is the graph associated to the $n \times n$ chessboard (a generalization of the classical $8 \times 8$ chessboard), with $n^2$ vertices, each one corresponding to a square of the chessboard. Two vertices of $\mathcal{Q}(n)$ are \textit{adjacent}, that is, linked by an edge, if and only if they are in the same row, in the same column or in the same diagonal of the chessboard. After a short overview on the main combinatorial properties of $\mathcal{Q}(n)$, its spectral properties are investigated. First, a lower bound on the least eigenvalue of an arbitrary graph is obtained using clique edge partitions and a sufficient condition for this lower bound be attained is deduced. For the particular case of $\mathcal{Q}(n)$, we prove that for every $n$, its least eigenvalue is not less than $-4$ and it is equal to $-4$ with multiplicity $(n-3)^2$, for every $n \ge 4$. Furthermore, $n-4$ is also an eigenvalue of $\mathcal{Q}(n)$, with multiplicity at least $\frac{n-2}{2}$ when $n$ is even and at least $\frac{n+1}{2}$ when $n$ is odd. A conjecture about the integer eigenvalues of $\mathcal{Q}(n)$ is presented. We finish this article with an algorithm to determine an equitable partition of the $n$-Queens' graph, $\mathcal{Q}(n)$, for $n \ge 3$, concluding that such equitable partition has $\frac{(\lceil n/2\rceil+1)\lceil n/2\rceil}{2}$ cells.