  Let E be an arbitrary directed graph with no restrictions on the number of
vertices and edges and let K be any field. We give necessary and sufficient
conditions for the Leavitt path algebra L_K(E) to be of countable irreducible
representation type, that is, we determine when L_K(E)has at most countably
many distinct isomorphism classes of simple left L_K(E-modules. It is also
shown that L_K(E) has dinitely many isomorphism classes of simple left modules
if and only if L_K(E) is a semi-artinian von Neumann regular ring with at most
finitely many ideals. Equivalent conditions on the graph E are also given.
Examples are constructed showing that for each (finite or infinite) cardinal m
there exists a Leavitt path algebra L having exactly m distinct isomorphism
classes of simple left modules.
