Research and Scholarship
*denotes a student collaborator
On Digraphs with Polygonal Restricted Numerical Range
In 2020, Cameron, Robertson, and I introduced the restricted numerical range of a digraph (directed graph) as a tool for characterizing digraphs and studying their algebraic connectivity. In particular, we showed that digraphs with a restricted numerical range of a single point, a horizontal line segment, and a vertical line segment were characterized as k-imploding stars, directed joins of bidirectional digraphs, and regular tournaments, respectively. In this article, we to extend these results by investigating digraphs whose restricted numerical range is a convex polygon in the complex plane. We provide computational methods for identifying these polygonal digraphs and show that these digraphs can be broken into three disjoint classes: normal, restricted-normal, and pseudo-normal digraphs, all of which are closed under the digraph complement. We prove sufficient conditions for normal digraphs and show that the directed join of two normal digraphs results in a restricted-normal digraph. Also, we prove that directed joins are the only restricted-normal digraphs when the order is square-free or twice a square-free number. Finally, we provide methods to construct restricted-normal digraphs that are not directed joins for all orders that are neither square-free nor twice a square-free number.
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.