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