On the restricted numerical range of the Laplacian matrix for digraphs

Cameron, Thomas R.; Robertson, M.; Wiedemann, Alexander

doi:10.1080/03081087.2020.1748853

Cited by 3 publications

(1 citation statement)

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“…The restricted numerical range is a novel tool for characterizing digraphs and studying their algebraic connectivity. In [2], digraphs with a restricted numerical range as a degenerate polygon, that is, a point or a line segment, are completely described. In this article, we extended these results to include digraphs whose restricted numerical range is a non-degenerate convex polygon in the complex plane.…”

Section: Discussionmentioning

confidence: 99%

On digraphs with polygonal restricted numerical range

Cameron¹,

Hall²,

Small³

et al. 2021

Preprint

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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.

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Section: Discussionmentioning

confidence: 99%