Classifying Leavitt path algebras up to involution preserving homotopy

We prove that the Bowen–Franks group classifies Leavitt path algebras of purely infinite simple finite graphs over a regular supercoherent commutative ring with involution where 2 is invertible, equipped their standard involutions, up to matricial stabilization and preserving homotopy equivalence. also consider twisting on obtain partial results in same direction for graphs. Our tools are K-theoretic, we several about (Hermitian, bivariant) K-theory algebras, such as Poincaré duality, which independent interest.