Notes on simplicial rook graphs

The simplicial rook graph 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 SR(m,n) has integral eigenvalues, and smallest eigenvalue s = max( -n, -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.