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 Morita equivalent to the crossed product C₀ ( ∂ T ) × G, where ∂ T is a certain zero-dimensional space canonically associated to the tree.