On the Spectra of Simplicial Rook Graphs

Abstract

The simplicial rook graph $SR(d,n)$ is the graph whose vertices are the lattice points in the $n$th dilate of the standard simplex in $\mathbb {R}^d$, with two vertices adjacent if they differ in exactly two coordinates. We prove that the adjacency and Laplacian matrices of $SR(3,n)$ have integral spectrum for every $n$. The proof proceeds by calculating an explicit eigenbasis. We conjecture that $SR(d,n)$ is integral for all $d$ and $n$, and present evidence in support of this conjecture. For $n<\left( {\begin{array}{c}d\\ 2\end{array}}\right)$, the evidence indicates that the smallest eigenvalue of the adjacency matrix is $-n$, and that the corresponding eigenspace has dimension given by the Mahonian numbers, which enumerate permutations by number of inversions.