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 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.

The Generalized Euler Characteristics of the Graphs Split at Vertices

“…In Refs. [21,22], the formulas for the Euler characteristic for graphs with the standard boundary conditions at the vertices and with the mixed ones, standard and Dirichlet boundary conditions at vertices, were derived. In the case of the standard boundary conditions,…”

“…From the experimental point of view, the usefulness of Equation ( 6) stems from the fact that the generalized Euler characteristic can be evaluated using only a limited number K = K min of the lowest eigenvalues (resonances) [21,22,68,69]…”

“…In our investigations, we used a set-up (see Figure 2) that consisted of an Agilent E8364B vector network analyzer (VNA) and HP 85133-60016 flexible microwave cable that connected the VNA to the measured network. The flexible cable connected to the network is equivalent to attaching an infinite lead to the quantum graph [22,32]. In this way, the one-port scattering matrix S 11 (ν) of the network was measured as a function of microwave frequency ν.…”

“…where |V| and |E| denote the number of vertices and edges of the graph. It is a purely topological quantity; however, it has been shown in [19][20][21][22] that it can also be defined by the graph and microwave network spectra. The formula describing the generalized Euler characteristic E [22,23], which is also applicable for graphs and networks with the Dirichlet boundary conditions, will be discussed later.…”

“…In this article, we will analyze the splitting of a quantum graph (network) into two disconnected subgraphs (subnetworks). Using a currently introduced spectral invariant-the generalized Euler characteristic E [22]-we determine the number |V c | of common vertices where the two subgraphs were initially connected. The application of the generalized Euler characteristic E for this purpose stems from the fact that it can be evaluated without knowing the topologies of quantum graphs (networks), using small or moderate numbers of their lowest eigenenergies (resonances).…”