Universal operator algebras of directed graphs

Given a directed graph, there exists a universal operator algebra and universal C*-algebra associated to the directed graph. For finite graphs this algebra decomposes as the universal free product of some building block operator algebras. For countable directed graphs, the universal operator algebras arise as direct limits of operator algebras of finite subgraphs. Finally, a method for computing the K-groups for universal operator algebras of directed graphs is given.