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On the Spectra of Simplicial Rook Graphs

The \emph{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 $n<\binom{d}{2}$, 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.