The negative tetrahedron and the first infinite family of connected digraphs that are strongly determined by the Hermitian spectrum
type: Item
instance of: scholarly article
title: The negative tetrahedron and the first infinite family of connected digraphs that are strongly determined by the Hermitian spectrum
zbMATH Open document ID: 1442.05078
author: Pepijn Wissing
author: Edwin R. Van Dam
published in: Journal of Combinatorial Theory. Series A
publication date: 2020-03-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/1903.09531
review text: It is very elusive to find digraphs that are uniquely determined by their Hermitian spectra. So, people usually turn to finding spectral determination of classes of switching equivalent digraphs, rather than individual digraphs. In the present paper, the authors return back to considering the traditional notion: the strongly determined digraph (or mixed graph) by its Hermitian spectrum (abbreviated SHDS), which is a digraph (or mixed graph) that is isomorphic to each digraph (or mixed graph) to which it is Hermitian cospectral. From numerical analysis (not a rigorous proof), they come to the conclusion that digraphs with this property are extremely rare. However, fortunately they can construct an infinite family of connected digraphs that is SHDS. This family is obtained via the introduction of twin vertices into a structure that is named negative tetrahedron. This special digraph, that exhibits extreme spectral behavior, is contained in the surprisingly small collection of all digraphs with exactly one negative eigenvalue, which is determined as an intermediate result.
Mathematics Subject Classification ID: 05C20
Mathematics Subject Classification ID: 05C50
Mathematics Subject Classification ID: 05C60
Mathematics Subject Classification ID: 05C40
zbMATH DE Number: 7180815
zbMATH Keywords: Hermitian adjacency matrix
zbMATH Keywords: spectra of digraphs
zbMATH Keywords: cospectral digraphs
zbMATH Keywords: spectral characterization of digraphs
reviewed by: Xueliang Li
MaRDI profile type: Publication
OpenAlex ID: W3006557542
arXiv ID: 1903.09531
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DOI: 10.1016/J.JCTA.2020.105232
description: scientific article