Exchange Leavitt path algebras and stable rank
  We characterize Leavitt path algebras which are exchange rings in terms of intrinsic properties of the graph and show that the values of the stable rank for these algebras are 1, 2 or $\infty$. Concrete criteria in terms of properties of the underlying graph are given for each case. 
G. Aranda-Pino, E. Pardo, M. Siles-Molina