Free subgroups in weighted Leavitt Path Algebras

We study unit groups of weighted Leavitt path algebras. Let $K$ be a field of characteristic $0$ and let $(E,\omega )$ be a finite connected weighted graph. We prove that $L_K(E,\omega {)}^{\times }$ is abelian if and only if $L_K(E,\omega )$ is a domain. Equivalently, $L_K(E,\omega {)}^{\times }$ contains no non-cyclic free subgroup if and only if $L_K(E,\omega )$ is a domain.