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On the spectra of simplicial rook graphs
Martin, Jeremy L. Wagner, Jennifer D.
Martin, Jeremy L.
Wagner, Jennifer D.
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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.
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2014-05-05
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Graph, Smplicial rook graph, Integral, Spectrum, Eigenvalues