A note on distance-regular graphs with a small number of vertices compared to the valency

In this note we study distance-regular graphs with a small number of vertices compared to the valency. We show that for a given α > 2, there are finitely many distance-regular graphs Γ with valency k, diameter D ≥ 3 and v vertices satisfying v≤αk unless (D = 3 and Γ is imprimitive) or (D = 4 and Γ is antipodal and bipartite). We also show, as a consequence of this result, that there are finitely many distance-regular graphs with valency k ≥ 3, diameter D ≥ 3 and c₂ ≥ εk for a given 0 < ε < 1 unless (D = 3 and Γ is imprimitive) or (D = 4 and Γ is antipodal and bipartite).