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Pseudo primitive idempotents and almost 2-homogeneous bipartite distance-regular graphs. (English) Zbl 1133.05105
The pseudo cosine sequence of a distance-regular graph \(\Gamma\) of diameter \(D\) is a sequence \(\sigma\) of scalars \(1=\sigma_0,\sigma_1,\dots,\sigma_D\) defined by a particular recurrence relation involving the intersection numbers of \(\Gamma\). The pseudo primitive idempotent \(E\) associated to \(\sigma\) is a non-0 scalar multiple of the matrix \(E=\sum_i \sigma_i A_i\), with \(A_i\)’s being the distance matrices of \(\Gamma\). It is shown that for a bipartite \(\Gamma\) of valency at least 3 and with \(D\geq 4\), and for \(\sigma\) satisfying \(| \sigma_1| \neq 1\) the following properties are equivalent: (i) \(E\circ E\) is a linear combination of all-ones matrix and a pseudo primitive idempotent; (ii) there exists \(\beta\) such that \(\beta\sigma_i=\sigma_{i-1}+\sigma_{i+1}\) for \(1\leq i \leq D-1\). Moreover, such \(\sigma\) and \(E\) exist if and only if \(\Gamma\) satisfies \(c_2\geq 2\) and is almost \(2\)-homogeneous.

MSC:
05E30 Association schemes, strongly regular graphs
05C75 Structural characterization of families of graphs
05C62 Graph representations (geometric and intersection representations, etc.)
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