  Given a directed graph, there exists a universal operator algebra and
universal C*-algebra associated to the directed graph. In this paper we give
intrinsic constructions of these objects. We provide an explicit construction
for the maximal C*-algebra of an operator algebra. We also discuss uniqueness
of the universal algebras for finite graphs, showing that for finite graphs the
graph is an isomorphism invariant for the universal operator algebra of a
directed graph. We show that the underlying undirected graph is a Banach
algebra isomorphism invariant for the universal C*-algebra of a directed graph.
