Show that a finite graph product of finitely generated groups is finitely generated, and similarly for finitely presented groups.
I think when we have a finitely generated groups,the graph product of it is contained something like a loop(condition on generators),every loop will be a generator for graph product,so it is finitely generated.
I know what I explain maybe sounds crazy but it was my feeling.please help me with your knowledge.thanks a lot.
The paper
R. Brown, M. Bullejos, T. Porter, , `Crossed complexes, free crossed resolutions and graph products of groups', Proceedings Workshop Korea 2000, Heldermann Verlag. 27 (2003) 11-26.
gives a crossed complex approach to this type of question on graph products, and also gives reference to related work. The paper is available here.