On digraphs with polygonal restricted numerical range

In 2020, Cameron et al. introduced the restricted numerical range of a digraph (directed graph) as a tool for characterizing digraphs and studying their algebraic connectivity. Notably, digraphs with a degenerate polygon (that is, a point or a line segment) as a restricted numerical range were completely described. In this article, we extend those results to include digraphs whose restricted numerical range is a non-degenerate convex polygon. In general, we refer to digraphs whose restricted numerical range is a degenerate or non-degenerate convex polygon as polygonal. We provide computational methods for identifying these polygonal digraphs and show that they can be broken into three disjoint classes: normal, restricted-normal, and pseudo-normal digraphs. Sufficient conditions for normal digraphs are provided, and we show that the directed join of two normal digraphs results in a restricted-normal digraph. Moreover, 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.