A characterization of Q-polynomial distance-regular graphs

Abstract

We obtain the following characterization of $Q$-polynomial distance-regular graphs. Let $\G$ denote a distance-regular graph with diameter $d\ge 3$. Let $E$ denote a minimal idempotent of $\G$ which is not the trivial idempotent $E_0$. Let $\{{\theta }_i^{\ast }{\}}_{i=0}^d$ denote the dual eigenvalue sequence for $E$. We show that $E$ is $Q$-polynomial if and only if (i) the entry-wise product $E\circ E$ is a linear combination of $E_0$, $E$, and at most one other minimal idempotent of $\G$; (ii) there exists a complex scalar $\beta$ such that ${\theta }_{i-1}^{\ast }-\beta {\theta }_i^{\ast }+{\theta }_{i+1}^{\ast }$ is independent of $i$ for $1\le i\le d-1$; (iii) ${\theta }_i^{\ast }\ne {\theta }_0^{\ast }$ for $1\le i\le d$.

10 pages, 1 figure