A new spectral invariant for quantum graphs

Abstract The Euler characteristic i.e., the difference between the number of vertices |V| and edges |E| is the most important topological characteristic of a graph. However, to describe spectral properties of differential equations with mixed Dirichlet and Neumann vertex conditions it is necessary t...

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Autores principales:	Michał Ławniczak, Pavel Kurasov, Szymon Bauch, Małgorzata Białous, Afshin Akhshani, Leszek Sirko
Formato:	article
Lenguaje:	EN
Publicado:	Nature Portfolio 2021
Materias:	Medicine R Science Q
Acceso en línea:	https://doaj.org/article/e0afc2d00ff44564adfd06cd42d8bc53
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spelling	oai:doaj.org-article:e0afc2d00ff44564adfd06cd42d8bc532021-12-02T16:06:43ZA new spectral invariant for quantum graphs10.1038/s41598-021-94331-02045-2322https://doaj.org/article/e0afc2d00ff44564adfd06cd42d8bc532021-07-01T00:00:00Zhttps://doi.org/10.1038/s41598-021-94331-0https://doaj.org/toc/2045-2322Abstract The Euler characteristic i.e., the difference between the number of vertices \|V\| and edges \|E\| is the most important topological characteristic of a graph. However, to describe spectral properties of differential equations with mixed Dirichlet and Neumann vertex conditions it is necessary to introduce a new spectral invariant, the generalized Euler characteristic $$\chi _G:= \|V\|-\|V_D\|-\|E\|$$ χ G : = \| V \| - \| V D \| - \| E \| , with $$\|V_D\|$$ \| V D \| denoting the number of Dirichlet vertices. We demonstrate theoretically and experimentally that the generalized Euler characteristic $$\chi _G$$ χ G of quantum graphs and microwave networks can be determined from small sets of lowest eigenfrequencies. If the topology of the graph is known, the generalized Euler characteristic $$\chi _G$$ χ G can be used to determine the number of Dirichlet vertices. That makes the generalized Euler characteristic $$\chi _G$$ χ G a new powerful tool for studying of physical systems modeled by differential equations on metric graphs including isoscattering and neural networks where both Neumann and Dirichlet boundary conditions occur.Michał ŁawniczakPavel KurasovSzymon BauchMałgorzata BiałousAfshin AkhshaniLeszek SirkoNature PortfolioarticleMedicineRScienceQENScientific Reports, Vol 11, Iss 1, Pp 1-9 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	Medicine R Science Q
spellingShingle	Medicine R Science Q Michał Ławniczak Pavel Kurasov Szymon Bauch Małgorzata Białous Afshin Akhshani Leszek Sirko A new spectral invariant for quantum graphs
description	Abstract The Euler characteristic i.e., the difference between the number of vertices \|V\| and edges \|E\| is the most important topological characteristic of a graph. However, to describe spectral properties of differential equations with mixed Dirichlet and Neumann vertex conditions it is necessary to introduce a new spectral invariant, the generalized Euler characteristic $$\chi _G:= \|V\|-\|V_D\|-\|E\|$$ χ G : = \| V \| - \| V D \| - \| E \| , with $$\|V_D\|$$ \| V D \| denoting the number of Dirichlet vertices. We demonstrate theoretically and experimentally that the generalized Euler characteristic $$\chi _G$$ χ G of quantum graphs and microwave networks can be determined from small sets of lowest eigenfrequencies. If the topology of the graph is known, the generalized Euler characteristic $$\chi _G$$ χ G can be used to determine the number of Dirichlet vertices. That makes the generalized Euler characteristic $$\chi _G$$ χ G a new powerful tool for studying of physical systems modeled by differential equations on metric graphs including isoscattering and neural networks where both Neumann and Dirichlet boundary conditions occur.
format	article
author	Michał Ławniczak Pavel Kurasov Szymon Bauch Małgorzata Białous Afshin Akhshani Leszek Sirko
author_facet	Michał Ławniczak Pavel Kurasov Szymon Bauch Małgorzata Białous Afshin Akhshani Leszek Sirko
author_sort	Michał Ławniczak
title	A new spectral invariant for quantum graphs
title_short	A new spectral invariant for quantum graphs
title_full	A new spectral invariant for quantum graphs
title_fullStr	A new spectral invariant for quantum graphs
title_full_unstemmed	A new spectral invariant for quantum graphs
title_sort	new spectral invariant for quantum graphs
publisher	Nature Portfolio
publishDate	2021
url	https://doaj.org/article/e0afc2d00ff44564adfd06cd42d8bc53
work_keys_str_mv	AT michałławniczak anewspectralinvariantforquantumgraphs AT pavelkurasov anewspectralinvariantforquantumgraphs AT szymonbauch anewspectralinvariantforquantumgraphs AT małgorzatabiałous anewspectralinvariantforquantumgraphs AT afshinakhshani anewspectralinvariantforquantumgraphs AT leszeksirko anewspectralinvariantforquantumgraphs AT michałławniczak newspectralinvariantforquantumgraphs AT pavelkurasov newspectralinvariantforquantumgraphs AT szymonbauch newspectralinvariantforquantumgraphs AT małgorzatabiałous newspectralinvariantforquantumgraphs AT afshinakhshani newspectralinvariantforquantumgraphs AT leszeksirko newspectralinvariantforquantumgraphs
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A new spectral invariant for quantum graphs

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