Notes on simplicial rook graphs

Abstract

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, - (m2)) , 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.