Homogeneity of a distance-regular graph which supports a spin model

A spin model is a square matrix that encodes the basic data for a statistical mechanical construction of link invariants due to V.F.R. Jones. Every spin model W is contained in a canonical Bose-Mesner algebra N (W). In this paper we study the distance-regular graphs Γ whose Bose-Mesner algebra M satisfies W ∈ M ⊆ N (W). Suppose W has at least three distinct entries. We show that Γ is 1-homogeneous and that the first and the last subconstituents of r are strongly regular and distance-regular, respectively.