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Bipartite distance-regular graphs: the \(Q\)-polynomial property and pseudo primitive idempotents. (English) Zbl 1297.05264
Summary: Let \(\Gamma\) denote a bipartite distance-regular graph with diameter at least 4 and valency at least 3. Fix a vertex of \(\Gamma\) and let \(T\) denote the corresponding Terwilliger algebra. Suppose that \(\Gamma\) is \(Q\)-polynomial and there are two non-isomorphic irreducible \(T\)-modules with endpoint 2. We show that, unless the intersection numbers of \(\Gamma\) fit one exceptional case (which is not known to correspond to an actual graph), the entry-wise product of pseudo primitive idempotents associated with these modules is a linear combination of two pseudo primitive idempotents. This result relates to a conjecture of MacLean and Terwilliger.

MSC:
05E30 Association schemes, strongly regular graphs
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