Abstract. We address the classi cation problem for graph C * -algebras of nite graphs ( nitely 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 uncountably many ideals.We nd 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. is proves in turn that under this condition, the graph C * -algebras are in fact classi able by K-theory, providing in particular complete classi cation when the C * -algebras in question are either of real rank zero or type I/postliminal. e 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 classi cation problem for the quantum lens spaces de ned by Hong and Szymański, and to complete the classi cation of graph C * -algebras associated to all simple graphs with four vertices or less.

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We give a classification result for a certain class of C * -algebras A over a finite topological space X in which there exists an open set U of X such that U separates the finite and infinite subquotients of A. We apply our results to C * -algebras arising from graphs.

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