A new spectral invariant for quantum graphs

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|-|VD|-|E|, with |VD| denoting the number of Dirichlet vertices. We demonstrate theoretically and experimentally that the generalized Euler characteristic chi 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 can be used to determine the number of Dirichlet vertices. That makes the generalized Euler characteristic chi 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.

Automated summary

The Euler characteristic and the generalized Euler characteristic are important for describing the topological and spectral properties of graphs with mixed vertex conditions. The generalized Euler characteristic can be determined from lowest eigenfrequencies and used to identify the number of Dirichlet vertices, making it a powerful tool for studying physical systems modeled by differential equations on metric graphs.