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Idea

Given two groups, two typical ways of combining them into a new group is by taking either their direct product or their free product. Given a finite family of groups, you could repeatedly take binary direct products to get the direct product of the whole family, or repeatedly take binary free products to get the free product of the whole family. One could if they wish take a mix of direct products and free products. Given a simple graph and a group assigned to each vertex, the graph product of this family of groups is a group which looks like a direct product if you look at two groups connected by an edge, and a free product otherwise.

References

The general notion of graph products of groups was first introduced in:

Elisabeth Green?, Graph products of groups, PhD thesis, University of Leeds, (1990). pdf

This was generalized to monoids in:

António Veloso da Costa?, Graph products of monoids, Semigroup Forum, Volume 63, Issue 2, pp 247–277 (2001). web

This was generalized to objects of pointed categories in: