Correspondences between graph C$^*$-algebras

We describe proper correspondences from graph C$^*$-algebras to arbitrary C$^*$-algebras by K-theoretic data. If the target C$^*$-algebra is a graph C$^*$-algebra as well, we may lift an isomorphism on a certain invariant to correspondences back and forth that are inverse to each other up to ideal-preserving homotopy. This implies a classification theorem for purely infinite graph C$^*$-algebras up to stable isomorphism.