  We compute the monoid $V(L_K(E))$ of isomorphism classes of finitely
generated projective modules over certain graph algebras $L_K(E)$, and we show
that this monoid satisfies the refinement property and separative cancellation.
We also show that there is a natural isomorphism between the lattice of graded
ideals of $L_K(E)$ and the lattice of order-ideals of $V(L_K(E))$. When $K$ is
the field $\mathbb C$ of complex numbers, the algebra $L_{\mathbb C}(E)$ is a
dense subalgebra of the graph $C^*$-algebra $C^*(E)$, and we show that the
inclusion map induces an isomorphism between the corresponding monoids. As a
consequence, the graph C*-algebra of any row-finite graph turns out to satisfy
the stable weak cancellation property.
