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Abstract

Abstract. Let denote a bipartite distance-regular graph with diameter D ≥ 4, valency k ≥ 3, and distinct eigenvalues θ 0 > θ 1 > · · · > θ D . Let M denote the Bose-Mesner algebra of . For 0 ≤ i ≤ D, let E i denote the primitive idempotent of M associated with θ i . We refer to E 0 and E D as the trivial idempotents of M. Let E, F denote primitive idempotents of M. We say the pair E, F is taut whenever (i) E, F are nontrivial, and (ii) the entry-wise product E • F is a linear combination of two distinct primitive idempotents of M. We show the pair E, F is taut if and only if there exist real scalars α, β such thatwhere σ 0 , σ 1 , . . . , σ D and ρ 0 , ρ 1 , . . . , ρ D denote the cosine sequences of E, F, respectively. We define to be taut whenever has at least one taut pair of primitive idempotents but is not 2-homogeneous in the sense of Nomura and Curtin. Assume is taut and D is odd, and assume the pair E, F is taut. We showUsing these equations, we recursively obtain σ 0 , σ 1 , . . . , σ D and ρ 0 , ρ 1 , . . . , ρ D in terms of the four real scalars σ, ρ, α, β. From this we obtain all intersection numbers of in terms of σ, ρ, α, β. We showed in an earlier paper that the pair E 1 , E d is taut, where d = (D − 1)/2. Applying our results to this pair, we obtain the intersection numbers of in terms of k, µ, θ 1 , θ d , where µ denotes the intersection number c 2 . We show that if is taut and D is odd, then is an antipodal 2-cover.