  We address the classification problem for graph $C^*$-algebras of finite
graphs (finitely many edges and vertices), containing the class of
Cuntz-Krieger algebras as a prominent special case. Contrasting earlier work,
we do not assume that the graphs satisfy the standard condition (K), so that
the graph $C^*$-algebras may come with uncountable ideal structures.
  We find that in this generality, stable isomorphism of graph $C^*$-algebras
does not coincide with the geometric notion of Cuntz move equivalence. However,
adding a modest condition on the graphs, the two notions are proved to be
mutually equivalent and equivalent to the $C^*$-algebras having isomorphic
$K$-theories. This proves in turn that under this condition, the graph
$C^*$-algebras are in fact classifiable by $K$-theory, providing in particular
complete classification when the $C^*$-algebras in question are either of real
rank zero or type I/postliminal. The key ingredient in obtaining these results
is a characterization of Cuntz move equivalence using the adjacency matrices of
the graphs.
  Our results are applied to discuss the classification problem for the quantum
lens spaces defined by Hong and Szyma\'nski, and to complete the classification
of graph $C^*$-algebras associated to all simple graphs with four vertices or
less.

Operator Algebras