Ideal-related $K$-theory for Leavitt path algebras and graph $C^*$-algebras Efren RuizMark Tomforde 46L3537B10graph C*-algebrasLeavitt path algebras We introduce a notion of ideal-related $K$-theory for rings, and use it to prove that if two complex Leavitt path algebras $L_{\mathbb{C}}(E)$ and $L_{\mathbb{C}}(F)$ are Morita equivalent (respectively, isomorphic), then the ideal-related $K$-theories (respectively, the unital ideal-related $K$-theories) of the corresponding graph $C^{*}$-algebras $C^{*}(E)$ and $C^{*}(F)$ 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 $L_{\mathbb{C}}(E)$ and $L_{\mathbb{C}}(F)$ implies strong Morita equivalence (respectively, isomorphism) of the graph $C^{*}$-algebras $C^{*}(E)$ and $C^{*}(F)$. 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. Indiana University Mathematics Journal 2013 text pdf 10.1512/iumj.2013.62.5123 10.1512/iumj.2013.62.5123 en Indiana Univ. Math. J. 62 (2013) 1587 - 1620 state-of-the-art mathematics http://iumj.org/access/