  We prove that if E and F are graphs with a finite number of vertices and an
infinite number of edges, if K is a field, and if L_K(E) and L_K(F) are simple
Leavitt path algebras, then L_K(E) is Morita equivalent to L_K(F) if and only
if K_0^\textnormal{alg} (L_K(E)) \cong K_0^\textnormal{alg} (L_K(F)) and the
graphs $E$ and $F$ have the same number of singular vertices, and moreover, in
this case one may transform the graph E into the graph F using basic moves that
preserve the Morita equivalence class of the associated Leavitt path algebra.
We also show that when K is a field with no free quotients, the condition that
E and F have the same number of singular vertices may be replaced by
K_1^\textnormal{alg} (L_K(E)) \cong K_1^\textnormal{alg} (L_K(F)), and we
produce examples showing this cannot be done in general. We describe how we can
combine our results with a classification result of Abrams, Louly, Pardo, and
Smith to get a nearly complete classification of unital simple Leavitt path
algebras - the only missing part is determining whether the "sign of the
determinant condition" is necessary in the finite graph case. We also consider
the Cuntz splice move on a graph and its effect on the associated Leavitt path
algebra.
