C*-algebras of directed graphs and group actions

Given a free action of a group G on a directed graph E we show that the crossed product of C*(E), the universal C*-algebra of E, by the induced action is strongly Morita equivalent to C*(E/G). Since every connected graph E may be expressed as the quotient of a tree T by an action of a free group G we may use our results to show that C*(E) is strongly Monta equivalent to the crossed product C0(∂T) × G, where ∂T is a certain zero-dimensional space canonically associated to the tree.