Distance-regular graphs with a relatively small eigenvalue multiplicity

Godsil showed that if Γ is a distance-regular graph with diameter D ≥ 3 and valency k ≥ 3, and θ is an eigenvalue of Γ with multiplicity m ≥ 2, then k ≤ (m+2) (m-2)/2. In this paper we will give a refined statement of this result. We show that if Γ is a distance-regular graph with diameter D ≥ 3, valency k ≥ 2 and an eigenvalue θ with multiplicity m ≥ 2, such that k is close to (m+2) (m-1)/2, then θ must be a tail. We also characterize the distance-regular graphs with diameter D ≥ 3, valency k ≥3 and an eigenvalue θ with multiplicity m ≥2 satisfying k = (m+2) (m-2)/2.