TY - JOUR
T1 - On digraphs with polygonal restricted numerical range
AU - Cameron, Thomas R.
AU - Hall, H. Tracy
AU - Small, Ben
AU - Wiedemann, Alexander
N1 - Funding Information:
This work was partially supported by the AMS-Simons Travel Grants, which are administered by the American Mathematical Society with support from the Simons Foundation .
Publisher Copyright:
© 2022 Elsevier Inc.
PY - 2022/6/1
Y1 - 2022/6/1
N2 - 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.
AB - 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.
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U2 - 10.1016/j.laa.2022.02.034
DO - 10.1016/j.laa.2022.02.034
M3 - Article
AN - SCOPUS:85125527259
SN - 0024-3795
VL - 642
SP - 285
EP - 310
JO - Linear Algebra and Its Applications
JF - Linear Algebra and Its Applications
ER -