A dynamical characterization of diagonal preserving $*$-isomorphisms of   graph $C^*$-algebras
  We characterize when there exists a diagonal preserving $*$-isomorphism between two graph $C^*$-algebras in terms of the dynamics of the boundary path spaces. In particular, we refine the notion of "orbit equivalence" between the boundary path spaces of the directed graphs $E$ and $F$ and show that this is a necessary and sufficient condition for the existence of a diagonal preserving $*$-isomorphism between the graph $C^*$-algebras $C^*(E)$ and $C^*(F)$. 
Sara E. Arklint and S{\o}ren Eilers and Efren Ruiz
math.OA