On the restricted numerical range of the Laplacian matrix for digraphs

In this article, we present the restricted numerical for the Laplacian matrix of a directed graph (digraph). We motivate our interest in the restricted numerical range by its close connection to the algebraic connectivity of a digraph. Moreover, we show that the restricted numerical range can be used to characterize digraphs, some of which are not determined by their Laplacian spectrum. Finally, we identify a new class of digraphs that are characterized by having a real restricted numerical range.