For a finite graph $X$, let $A\_X$ denote the associated right-angled Artin group. Thus $A\_X$ is generated by the vertices of $X$ subject to the relations $[v,w]=1$ whenever vertices $v$ and $w$ are connected by an edge.


I have seen references to the following theorem in several places, but I can neither figure out a proof myself not find a reference that proves it:


Let $X$ be a finite graph. Assume that $A\_X \cong G \times H$, where $G$ and $H$ are nontrivial groups. Then $X$ decomposes as a nontrivial join, i.e., we can partition the vertices of $X$ into two nonempty sets $V\_1$ and $V\_2$ such that each vertex in $V\_1$ is connected by an edge to each vertex in $V\_2$ and also each vertex in $V\_2$ is connected by an edge to each vertex in $V\_1$.


Can anyone provide me a proof or a reference for this?



Right-angled Artin groups that split as direct products
The place I've seen this is in Koberda's RAAG notes [here](https://users.math.yale.edu/users/koberda/raagcourse.pdf), see Corollary 2.15. This relies on the description of centralizers in Proposition 2.14, which is also proved in Behrstock and Charney's paper [here](https://link.springer.com/article/10.1007/s00208-011-0641-8) (if you want an officially-published reference). Given Proposition 2.14, Corollary 2.15 is immediate because if $A\_X\cong G\times H$ for non-trivial $G$ and $H$ then every non-trivial element $g$ has non-cyclic centralizer, so picking some $g$ that "really uses" every vertex of $X$, Proposition 2.14 implies $X$ is a non-trivial join.


