  Naimark's problem asks whether a C*-algebra that has only one irreducible
*-representation up to unitary equivalence is isomorphic to the C*-algebra of
compact operators on some (not necessarily separable) Hilbert space. This
problem has been solved in special cases, including separable C*-algebras and
Type I C*-algebras. However, in 2004 Akemann and Weaver used the diamond
principle to construct a C*-algebra with $\aleph_1$ generators that is a
counterexample to Naimark's Problem. More precisely, they showed that the
statement "There exists a counterexample to Naimark's Problem that is generated
by $\aleph_1$ elements." is independent of the axioms of ZFC. Whether Naimark's
problem itself is independent of ZFC remains unknown. In this paper we examine
Naimark's problem in the setting of graph C*-algebras, and show that it has an
affirmative answer for (not necessarily separable) AF graph C*-algebras as well
as for C*-algebras of graphs in which each vertex emits a countable number of
edges.
