  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 of the associated extension 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, infinite collections of such sequences comprise 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 $E$.
