  Leavitt path algebras are associated to di(rected )graphs and there is a
combinatorial procedure (the reduction algorithm) making the digraph smaller
while preserving the Morita type. We can recover the vertices and most of the
arrows of the completely reduced digraph from the module category of a Leavitt
path algebra of polynomial growth. We give an explicit classification of all
irreducible representations of when the coefficients are a commutative ring
with 1. We define a Morita invariant filtration of the module category by Serre
subcategories and as a consequence we obtain a Morita invariant (the weighted
Hasse diagram of the digraph) which captures the poset of the sinks and the
cycles of $\Gamma$, the Gelfand-Kirillov dimension and more. When the
Gelfand-Kirillov dimension of the Leavitt path algebra is less than 4, the
weighted Hasse diagram (equivalently, the complete reduction of the digraph) is
a complete Morita invariant.
