Index maps in the K-theory of graph algebras

Let C*(E) be the graph C*-algebra associated to a graph E and let J be a gauge-invariant ideal in C*(E). We compute the cyclic six-term exact sequence in K-theory associated to the extension 0 -> J -> C* (E) -> C* (E)/J -> 0 in terms of the adjacency matrix associated to E. The ordered six-term exact sequence is a complete stable isomorphism invariant for several classes of graph C*-algebras, for instance those containing a unique proper nontrivial ideal. Further, in many other cases, finite collections of such sequences constitute complete invariants. Our results allow for explicit computation of the invariant, giving an exact sequence in terms of kernels and cokernels of matrices determined by the vertex matrix of E.