  We introduce a notion of ideal-related K-theory for rings, and use it to
prove that if two complex Leavitt path algebras are Morita equivalent
(respectively, isomorphic), then the ideal-related K-theories (respectively,
the unital ideal-related K-theories) of the corresponding graph C*-algebras are
isomorphic. This has consequences for the "Morita equivalence conjecture" and
"isomorphism conjecture" for graph algebras, and allows us to prove that when E
and F belong to specific collections of graphs whose C*-algebras are classified
by ideal-related K-theory, Morita equivalence (respectively, isomorphism) of
the Leavitt path algebras implies strong Morita equivalence (respectively,
isomorphism) of the graph C*-algebras. We state a number of corollaries that
describe various classes of graphs where these implications hold. In addition,
we conclude with a classification of Leavitt path algebras of amplified graphs
similar to the existing classification for graph C*-algebras of amplified
graphs.
