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arXiv · 2407.00714

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

Mojtaba Jazaeri
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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.

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Mojtaba Jazaeri. 2025-03-22. On $Q$-polynomial distance-regular graphs with a linear dependency involving a $3$-clique. https://arxiv.org/abs/2407.00714

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