On digraphs with polygonal restricted numerical range

TL;DR

This paper extends the concept of restricted numerical range of digraphs to convex polygons in the complex plane, providing classification, computational methods, and structural results for these polygonal digraphs.

Contribution

It introduces a classification of polygonal restricted numerical range digraphs into normal, restricted-normal, and pseudo-normal classes, with new structural theorems and construction methods.

Findings

Polygonal restricted numerical range digraphs are classified into three classes.

Directed joins of normal digraphs produce restricted-normal digraphs.

Construction methods for restricted-normal digraphs outside directed joins are provided.

Abstract

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. In particular, 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 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.