Taut Distance-Regular Graphs of Odd Diameter

Let denote a bipartite distance-regular graph with diameter D 4, valency k 3, and distinct eigenvalues ₀ > ₁ > ··· > _D. Let M denote the Bose-Mesner algebra of . For 0 i D, let E _i denote the primitive idempotent of M associated with _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 , such that _{i + 1 i + 1} – _{i – 1 i – 1} = _i ( _{i + 1} – _{i – 1}) + _i ( _{i + 1} – _{i – 1}) + (1 i D – 1)where ₀, ₁, ..., _D and ₀, ₁, ..., _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 showfor 1 i D – 1, where = ₁, = ₁. Using these equations, we recursively obtain ₀, ₁, ..., _D and ₀, ₁, ..., _D in terms of the four real scalars , , , . From this we obtain all intersection numbers of in terms of , , , . 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, , ₁, _d, where denotes the intersection number c ₂. We show that if is taut and D is odd, then is an antipodal 2-cover.