A characterization of Q-polynomial distance-regular graphs

TL;DR

This paper provides a new characterization of Q-polynomial distance-regular graphs using algebraic and spectral properties of their minimal idempotents and dual eigenvalues, offering a precise criterion for identifying such graphs.

Contribution

It introduces a novel characterization of Q-polynomial property in distance-regular graphs based on minimal idempotents and dual eigenvalues, extending existing theoretical understanding.

Findings

E is Q-polynomial iff E ∘ E is a linear combination of E0, E, and at most one other minimal idempotent.

Existence of a scalar β such that the dual eigenvalues satisfy a three-term recurrence.

Dual eigenvalues θ*_i are distinct from θ*_0 for all i ≥ 1.

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 $\{\theta_i^*\}_{i=0}^d$ 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 $\theta^*_{i-1}-\beta \theta^*_i + \theta^*_{i+1}$ is independent of $i$ for $1 \le i \le d-1$; (iii) $\theta^*_i \ne \theta^*_0$ for $1 \le i \le d$.