On the Spectra of Simplicial Rook Graphs

TL;DR

This paper investigates the spectral properties of simplicial rook graphs, proving integrality of spectra for SR(3,n) and conjecturing a generalization to all dimensions and sizes, supported by combinatorial evidence.

Contribution

It provides an explicit eigenbasis for SR(3,n) and conjectures integrality of spectra for all SR(d,n), advancing understanding of their algebraic structure.

Findings

The adjacency and Laplacian spectra of SR(3,n) are integral for all n.

For n<binomial(d,2), the smallest eigenvalue is -n with eigenspace dimension given by Mahonian numbers.

Evidence suggests the smallest eigenvalue of SR(d,n) is -n and the eigenspace dimension relates to permutation inversions.

Abstract

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.