Abstract: For a given an oriented graph E and a field $\ell$ a Leavitt path algebra LE over $\ell$ is defined. When $\ell$ is the complex numbers, LE carries a C*-norm, whose completion is the graph C*-algebra C*(E). An old theorem of Cuntz and R\ordam (predating the Kirchberg-Phillips theorem) says that if E and F are finite graphs such that C*(E) and C*(F) are purely infinite simple, then K_0(C*(E)) and K_0(C*(F)) are isomorphic as scaled order groups, then C*(E) and C*(F) are isomorphic. The proof uses Kasparov's K-theory and a result of Elliot on aproximately unitary equivalence. We remark that K_0(C*(E))=K_0(LE) for any graph E and any field \ell, and that LE is purely infinite simple precisely when C*(E) is. However it is not obvious how to adapt the proof (particularly the aproximately unitary part of it) in the purely algebraic setting, and the question of whether the same result holds for Leavitt path algebras is still open. In the talk I will present recent joint work with my student Diego Montero, which shows that, if E and F are finite and LE and LF are purely infinite simple, then any isomorphism K_0(LE)\cong K_0(LF) of scaled ordered groups lifts to a unital algebraic (i.e. polynomial) homotopy equivalence LE\to LF. The proof uses algebraic bivariant K-theory.