  We characterise when the Leavitt path algebras over $\mathbb{Z}$ of two
arbitrary countable directed graphs are $*$-isomorphic by showing that two
Leavitt path algebras over $\mathbb{Z}$ are $*$-isomorphic if and only if the
corresponding graph groupoids are isomorphic (if and only if there is a
diagonal preserving isomorphism between the corresponding graph
$C^*$-algebras). We also prove that any $*$-homomorphism between two Leavitt
path algebras over $\mathbb{Z}$ maps the diagonal to the diagonal. Both results
hold for slight more general subrings of $\mathbb{C}$ than just $\mathbb{Z}$.
