Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: ao-sci
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The categories of this paper are: atom-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: mtrl-th
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-lat
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The categories of this paper are: astro-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: nlin
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The categories of this paper are: nucl-th
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The categories of this paper are: supr-con
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The categories of this paper are: funct-an
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The categories of this paper are: comp-gas
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The categories of this paper are: eess
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The categories of this paper are: solv-int
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The categories of this paper are: nucl-ex