Graph product of groups

Definition
Let $$X = (V,E)$$ be a graph. To each $$v$$ in $$V$$, associate a vertex group $$G_v$$. Then, the graph product of the $$G_v$$s with respect to the graph $$X$$ is defined as $$F/R$$ where $$F$$ is the free product of all the $$G_v$$s and $$R$$ is the normal subgroup generated by subgroups of the form $$[G_u,G_v]$$ whenever there is an edge joining $$u$$ and $$v$$.

Direct product
The external direct product (restricted version) of a family of groups is precisely the same as the graph product with these as the vertex groups and the graph as the clique.

Free product
The free product of a family of groups is precisely the same as the graph product with these as the vertex groups and the graph taken as the empty graph.