  To every directed graph $E$ one can associate a \emph{graph inverse
semigroup} $G(E)$, where elements roughly correspond to possible paths in $E$.
These semigroups generalize polycylic monoids, and they arise in the study of
Leavitt path algebras, Cohn path algebras, Cuntz-Krieger $C^*$-algebras, and
Toeplitz $C^*$-algebras. We investigate topologies that turn $G(E)$ into a
topological semigroup. For instance, we show that in any such topology that is
Hausdorff, $G(E)\setminus \{0\}$ must be discrete for any directed graph $E$.
On the other hand, $G(E)$ need not be discrete in a Hausdorff semigroup
topology, and for certain graphs $E$, $G(E)$ admits a $T_1$ semigroup topology
in which $G(E)\setminus \{0\}$ is not discrete. We also describe, in various
situations, the algebraic structure and possible cardinality of the closure of
$G(E)$ in larger topological semigroups.
