Given vectors $V$ of length $d$, construct a graph $G = (V, E)$ where ${u, v} \in E$ iff the Pearson correlation between $u$ and $v$ is larger than some threshold $t > 0$. Is $G$ chordal? It seems like it should be, because a long chordless cycle like $a$ correlates with $b$, $b$ with $c$, $c$ with $d$ but nothing else seems difficult to construct. However, I cannot find a simple proof or a reference.
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It was described in this previous question how to obtain a correlation matrix whose entries come from the scalar product of certain vectors $u_1, u_2, \dots,u_n$. If we let the vectors be $$u_i=(1, \cos(\frac{2\pi i}{n}), \sin(\frac{2\pi i}{n}),0,\dots, 0)$$ we can set a high enough threshold so that the corresponding graph is a cycle of length $n$ and thus not chordal. |
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