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.

The Euler characteristic chi = vertical bar V vertical bar - vertical bar E vertical bar is an important topological characteristic of graphs and networks. Here, vertical bar V vertical bar and vertical bar E vertical bar denote the number of vertices and edges of a graph or a network. It has been shown in [Phys. Rev. E 101, 052320 (2020)] that the Euler characteristic can be determined from a finite sequence of the lowest eigenenergies lambda(1), ..., lambda(N) of a simple quantum graph. We will test this finding numerically, using chaotic graphs with vertical bar V vertical bar = 8 vertices. We will consider complete (fully connected) and incomplete realizations of 8-vertex graphs. The properties of the Euler characteristic will also be tested experimentally using the sequence of the lowest resonances of the 5-vertex microwave network. We will show that the Euler characteristic chi can be used to reveal whether the graph is planar or not.

The Euler characteristic chi = vertical bar V vertical bar - vertical bar E vertical bar and the total length L are the most important topological and geometrical characteristics of a metric graph. Here vertical bar V vertical bar and vertical bar E vertical bar denote the number of vertices and edges of a graph. The Euler characteristic determines the number beta of independent cycles in a graph while the total length determines the asymptotic behavior of the energy eigenvalues via Weyl's law. We show theoretically and confirm experimentally that the Euler characteristic can be determined (heard) from a finite sequence of the lowest eigenenergies lambda(1 ), ..., lambda(N) of a simple quantum graph, without any need to inspect the system visually. In the experiment quantum graphs are simulated by microwave networks. We demonstrate that the sequence of the lowest resonances of microwave networks with beta <= 3 can be directly used in determining whether a network is planar, i.e., can be embedded in the plane. Moreover, we show that the measured Euler characteristic chi can be used as a sensitive revealer of the fully connected graphs.