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For a finite graph $\Gamma$, its Right-angled Artin group isIts Salvetti complex $S_{\Gamma }$ is the one-vertex cube complex with an oriented loop labelled $v$ for each vertex of $\Gamma$, a square torus for every edge, and more generally one cubulated $k$-torus for every $k$-clique, attached compatibly along coordinate subtori. Thus ${\pi }_1(S_{\Gamma })\cong A_{\Gamma }$. The vertex link of a Salvetti complex is flag, so $S_{\Gamma }$ is a nonpositively curved cube complex.

For each $v\in V(\Gamma )$, the vertex link of a Salvetti complex has two vertices $v^{+}$ and $v^{-}$. If $v$ has degree $d_{\Gamma }(v)$, then each of $v^{+}$ and $v^{-}$ is adjacent to both signed vertices $w^{+},w^{-}$ for every neighbour $w$, and hence has link degree $2d_{\Gamma }(v)$.

Suppose the whole link is a cycle. Every link vertex has degree two, so every vertex of $\Gamma$ has degree one. The link is connected, which forces $\Gamma$ to be connected. A connected one-regular graph consists of one edge. Conversely, if $\Gamma$ is one edge, then $S_{\Gamma }$ is the square torus and its unique vertex has link the four-cycleThus the vertex link is a cycle exactly when $\Gamma$ is an edge.