  We prove that the C*-algebra of a directed graph $E$ is liminal iff the graph
satisfies the finiteness condition: if $p$ is an infinite path or a path ending
with a sink or an infinite emitter, and if $v$ is any vertex, then there are
only finitely many paths starting with $v$ and ending with a vertex in $p$.
Moreover, C*(E) is Type I precisely when the circuits of $E$ are either
terminal or transitory, i.e., $E$ has no vertex which is on multiple circuits,
and $E$ satisfies the weaker condition: for any infinite path $\lambda$, there
are only finitely many vertices of $\lambda$ that get back to $\lambda$ in an
infinite number of ways.

Operator Algebras