  We study path rings, Cohn path rings, and Leavitt path rings associated to
directed graphs, with coefficients in an arbitrary ring $R$. For each of these
types of rings, we stipulate conditions on the graph that are necessary and
sufficient to ensure that the ring satisfies either the rank condition or the
strong rank condition whenever $R$ enjoys the same property. In addition, we
apply our result for path rings and the strong rank condition to characterize
the graphs that give rise to amenable path algebras and exhaustively amenable
path algebras.
