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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 \(\varGamma \).
05E30 Association schemes, strongly regular graphs
05C12 Distance in graphs
Full Text: DOI
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