On $Q$-polynomial distance-regular graphs with a linear dependency involving a $3$-clique

TL;DR

The paper classifies $Q$-polynomial distance-regular graphs in which a $3$-clique $\{x,y,z\}$ yields a linearly dependent triple $E\hat{x},E\hat{y},E\hat{z}$. It shows this condition forces the cosine sequence to be $\{\sigma_i\}=(-\tfrac{1}{2})^i$, the corresponding eigenvalue to be the minimal $-\tfrac{k}{2}$, and $a_1=1$, with $a_i=c_i$. The graphs fall into explicit families: (i) those with classical parameters $(D,-2,\alpha,2+\alpha-\alpha[_{1}^{D}])$; (ii) regular near $2D$-gons of order $(2,t)$; or (iii) a finite list including four small regular near polygons and the dual polar graph $A_{2D-1}(2)$. This yields a complete classification under the stated linear-dependence condition and connects to Terwilliger’s framework for $Q$-polynomial graphs.

Abstract

Let $Γ$ denote a distance-regular graph with diameter $D \geq 2$. Let $E$ denote a primitive idempotent of $Γ$ with respect to which $Γ$ is $Q$-polynomial. Assume that there exists a $3$-clique $\{x,y,z\}$ such that $E\hat{x},E\hat{y},E\hat{z}$ are linearly dependent. In this paper, we classify all the $Q$-polynomial distance-regular graphs $Γ$ with the above property. We describe these graphs from multiple points of view.

Theorems & Definitions (24)

• Theorem 1.2
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Definition 3.3
• Definition 3.4
• Proposition 3.5
• proof
• Definition 3.6