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 6 (m+2)(m−1) 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−1) 2 .