Distance-regular graphs with a relatively small eigenvalue multiplicity

Abstract

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