  Distinguishing sets of quantum states shared by two parties using only local
operations and classical communication measurements is a fundamental topic in
quantum communication and quantum information theory. We introduce a
graph-theoretic approach, based on the theory of vector representations of
graphs, to the core problem of distinguishing product states with one-way LOCC.
We establish a number of results that show how distinguishing such states can
be framed in terms of properties of the underlying graphs associated with a set
of vector product states. We also present a number of illustrative examples.
