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In this paper we study the elementary theory of graph products of groups andshow that under natural conditions on the vertex groups we can recover (thecore of) the underlying graph and the associated vertex groups. More precisely,we require the vertex groups to satisfy a non-generic almost positive sentence,a condition which generalizes a range of natural ``non-freeness conditions"such as the satisfaction of a group law, having nontrivial center or beingboundedly simple.  As a corollary, we determine an invariant of the elementary theory of aright-angled Artin group, the core of the defining graph, which we conjectureto determine the elementary class of the RAAG. We further combine our resultswith the results of Sela on free products of groups to describe all finitelygenerated groups elementarily equivalent to certain RAAGs. We also deducerigidity results on the elementary classification of graph products of groupsfor specific types of vertex groups, such as finite, nilpotent or classicallinear groups.

On the elementary theory of graph products of groups