Bounding the diameter of a distance regular graph by a function ofkd

LetΓ be a distance regular graph with diameterd, andΓd(α) the set of vertices at distanced fromα. Γ is said to be thin if the induced subgraph onΓd(α) is a union of cliques for every vertexα. We show that the diameterd is bounded above by a function depending only onkd, which is the cardinality ofΓd(α), ifΓ is not thin. We also investigate thin distance regular graphs withad ≠ 0.