an:01672512
Zbl 0993.05148
Tomiyama, Masato
On the primitive idempotents of distance-regular graphs
EN
Discrete Math. 240, No. 1-3, 281-294 (2001).
00079680
2001
j
05E30 05C12 05C50
distance-regular graph; primitive idempotents; eigenvalue; cosines; \(Q\)-polynomial
From the author's abstract: Let \(\Gamma\) denote a distance-regular graph with diameter \(d\geq 3\). Let \(E\), \(F\) denote nontrivial primitive idempotents of \(\Gamma\) such that \(F\) corresponds to the second largest or least eigenvalue. We investigate the situation 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 inequalities involving the cosines of \(E\), and to show that equality is closely related to \(\Gamma\) being \(Q\)-polynomial with respect to \(E\). This generalizes a result of Lang on bipartite graphs and a result of Pascasio on tight graphs.
R.E.L.Aldred (Dunedin)