Notes on simplicial rook graphs

Abstract

The simplicial rook graph $\mathrm{SR}(m,n)$ is the graph of which the vertices are the sequences of nonnegative integers of length $m$ summing to $n$, where two such sequences are adjacent when they differ in precisely two places. We show that $\mathrm{SR}(m,n)$ has integral eigenvalues, and smallest eigenvalue $s=\max (-n,-(\genfrac{}{}{0ex}{}{m}{2}))$, and that this graph has a large part of its spectrum in common with the Johnson graph $J(m+n-1,n)$. We determine the automorphism group and several other properties.