Regular weights and the Johnson scheme

Abstract

This work addresses existence of weighted coherent configurations (cc’s) in the special case of the Johnson schemes,J(n, k). Many examples of weighted cc’s occur in the context of a permutation representation of a group which affords the cc. We determine the ranks of the weighted Johnson schemes which arise in this “group case” in connection with the alternating groupAn. The association schemeJ(n, 2) has rank 3, with the triangular graphT(n) as one of its graphs. Conditions on feasible parameters for regular, full rank weights onT(n) are established. We describe two infinite families of such feasible parameters.