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Weakly \(Q\)-polynomial distance-regular graphs. (English) Zbl 0935.05101
This note is based on the author’s paper [A note on the primitive idempotents of distance-regular graphs (preprint)]. In the first section, we state our results. In the next section, we review some definitions and basic concepts. For more background information, the reader may refer to E. Bannai and T. Itô [Algebraic combinatorics. I: Association schemes (1984; Zbl 0555.05019)], A. E. Brouwer, A. M. Cohen and A. Neumaier [Distance-regular graphs (1989; Zbl 0747.05073)] or C. D. Godsil [Algebraic combinatorics (1993; Zbl 0784.05001)].
Let \(\Gamma\) denote a distance-regular graph with diameter \(d\geq 3\) and eigenvalues \(\theta_0>\theta_1>\cdots> \theta_d\). Let \(E\) and \(F\) denote nontrivial primitive idempotents of \(\Gamma\). In [Tight graphs and their primitive idempotents, J. Algebr. Conb. 10, No. 1, 47-59 (1999; Zbl 0927.05085)], A. A. Pascasio investigated the situation that \(E\circ F\) is a scalar multiple of a primitive idempotent \(H\) of \(\Gamma\). She showed this occurs exactly when \(\Gamma\) is either bipartite or tight (in the sense of A. Jurišić, J. Koolen and P. M. Terwilliger [Tight distance-regular graphs (preprint)]). Moreover, she showed that at least one of \(E\) and \(F\) is equal to \(E_d\) if \(\Gamma\) is bipartite, and that \(E\) and \(F\) are a permutation of \(E_1\) and \(E_d\) if \(\Gamma\) is tight. If \(\Gamma\) is bipartite, M. Lang obtained an inequality involving the cosines of \(E\), and showed that equality is closely related to \(\Gamma\) being \(Q\)-polynomial with respect to \(E\). See [M. Lang, A new inequality for bipartite distance-regular graphs (preprint)]. If \(\Gamma\) is tight, A. A. Pascasio obtained similar inequalities involving the cosines of \(E\), and showed that again equality is closely related to \(\Gamma\) being \(Q\)-polynomial with respect to \(E\). See [A. A. Pascasio, Tight distance-regular graphs and the \(Q\)-polynomial property (preprint)].
In this note, we investigate a slightly more general situation. Let \(E\) denote a nontrivial primitive idempotent of \(\Gamma\) and let \(F\in \{E_1,E_d\}\). Our situation is that there exists a primitive idempotent \(H\) of \(\Gamma\) such that \(E\circ F\) is a linear combination of \(F\) and \(H\). Our main purpose is to obtain the above inequalities under our general assumption, and to show that again equality is closely related to \(\Gamma\) being \(Q\)-polynomial with respect to \(E\).
05E30 Association schemes, strongly regular graphs