Scientific Reports (Jul 2021)

A new spectral invariant for quantum graphs

  • Michał Ławniczak,
  • Pavel Kurasov,
  • Szymon Bauch,
  • Małgorzata Białous,
  • Afshin Akhshani,
  • Leszek Sirko

DOI
https://doi.org/10.1038/s41598-021-94331-0
Journal volume & issue
Vol. 11, no. 1
pp. 1 – 9

Abstract

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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.