$K$-theory of $C^*$-algebras of directed graphs

For a directed graph $E$, we compute the $K$-theory of the $C^*$-algebra $C^*(E)$ from the Cuntz-Krieger generators and relations. First we compute the $K$-theory of the crossed product $C^*(E)\times_\gamma\IT$, and then using duality and the Pimsner-Voiculescu exact sequence we compute the $K$-theory of $C^*(E)\otimes\CK \cong (C^*(E)\times\IT)\times\IZ$. The method relies on the decomposition of $C^*(E)$ as an inductive limit of Toeplitz graph $C^*$-algebras, indexed by the finite subgraphs of $E$. The proof and result require no special asssumptions about the graph, and is given in graph-theoretic terms. This can be helpful if the graph is described by pictures rather than by a matrix.