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Bipartite distance-regular graphs and taut pairs of pseudo primitive idempotents. (English) Zbl 1417.05250
Summary: Let \(\varGamma \) denote a bipartite distance-regular graph with diameter \(D \ge 4\), valency \(k \ge 3\), and intersection numbers \(c_i\), \(b_i \; (0\le i\le D)\). By a pseudo cosine sequence of \(\varGamma \) we mean a sequence of complex scalars \(\sigma _0, \sigma _1, \ldots , \sigma _D\) such that \(\sigma _0=1\) and \(c_i \sigma _{i-1} +b_i \sigma _{i+1} =k \sigma _1 \sigma _i\) for \(1\le i\le{D-1}\). By an associated pseudo primitive idempotent of \(\varGamma \), we mean a nonzero scalar multiple of the matrix \(\sum _{i=0}^D \sigma _i A_i\), where \(A_0, A_1, \ldots , A_D\) are the distance matrices of \(\varGamma \). Given pseudo primitive idempotents \(E\), \(F\) of \(\varGamma \), we define the pair \(E, F\) to be taut whenever the entry-wise product \(E \circ F\) is not a scalar multiple of a pseudo primitive idempotent, but is a linear combination of two pseudo primitive idempotents of \(\varGamma \). In this paper, we determine all the taut pairs of pseudo primitive idempotents of \(\varGamma \).
MSC:
05E30 Association schemes, strongly regular graphs
05C12 Distance in graphs
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