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.