  In this article, we extend the notion of the Laplacian spread to simple
directed graphs (digraphs) using the restricted numerical range. First, we
provide Laplacian spread values for several families of digraphs. Then, we
prove sharp upper bounds on the Laplacian spread for all polygonal and balanced
digraphs. In particular, we show that the validity of the Laplacian spread
bound for balanced digraphs is equivalent to the Laplacian spread conjecture
for simple undirected graphs, which was conjectured in 2011 and proven in 2021.
Moreover, we prove an equivalent statement for weighted balanced digraphs with
weights between $0$ and $1$. Finally, we state several open conjectures that
are motivated by empirical data.
