Abstract
Let Λ be a distance-regular graph of diameter d and valency k >2. If b t =1 and 2 t ⩽ d , then Λ is an antipodal double-cover. Consequently, if f >2 is the multiplicity of an eigenvalue of the adjacency matrix of Λ and if Λ is not an antipodal double-cover then d ⩽2 f −3. This result is an improvement of Godsil's bound.
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