Algebraic bivariant $K$-theory and Leavitt path algebras.

We investigate to what extent homotopy invariant, excisive and matrix stable homology theories help one distinguish between the Leavitt path algebras $L(E)$ $L(F)$ of graphs $E$ $F$ over a commutative ground ring $\ell$. approach this by studying structure such under bivariant algebraic $K$-theory $kk$, which is universal theory with properties above. show that very mild assumptions on $\ell$, for graph finitely many vertices reduced incidence $A_E$, in $kk$ depends only groups Coker$(I-A_E)$ Coker$(I-A_E^t)$. also prove algebras, has several similar those Kasparov's $C^*$-graph including analogues Universal coefficient Künneth theorems Rosenberg Schochet.