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>2α>2, there are finitely many distance-regular graphs ΓΓ with valency kk, diameter D≥3D≥3 and vv vertices satisfying v≤αkv≤αk unless (D=3D=3 and ΓΓ is imprimitive) or (D=4D=4 and ΓΓ is antipodal and bipartite). We also show, as a consequence of this result, that there are finitely many distance-regular graphs with valency k≥3k≥3, diameter D≥3D≥3 and c2≥εkc2≥εk for a given 0<ε<10<ε<1 unless (D=3D=3 and ΓΓ is imprimitive) or (D=4D=4 and ΓΓ is antipodal and bipartite).