  We establish necessary and sufficient conditions on a (not necessarily
countable) graph E for the graph C*-algebra C*(E) to be primitive. Along with a
known characterization of the graphs E for which C*(E) is prime, our main
result provides us with a systematic method for easily producing large classes
of (necessarily nonseparable) C*-algebras that are prime but not primitive. We
also compare and contrast our results with similar results for Leavitt path
algebras.
