**1. Introduction**

The concept of graphs was already introduced in the XVIII century by Leonhard Euler [1]. Two hundred years later, Linus Pauling [2] considered quantum graphs in order to describe the motion of quantum particles in a physical network. The models of quantum graphs were widely used to investigate many physical systems, e.g., quantum wires [3], mesoscopic quantum systems [4,5], a topological edge invariant [6], and the photon number statistics of coherent light [7]. Broad applications of graphs and networks mean that the theory of quantum graphs has been a subject of extensive research [8–14].

We will consider a metric graph Γ = (*V*, *E*), which consists of *v* vertices, *v* ∈ *V*, connected by *<sup>e</sup>* edges, *<sup>e</sup>* <sup>∈</sup> *<sup>E</sup>*. The edges *<sup>e</sup>* are intervals of the length *le* on the real line <sup>R</sup>. The metric graph becomes quantum when we equip it with the free Schrödinger operator. In our case, this is the one-dimensional Laplace operator, which equals *<sup>L</sup>*(Γ) = <sup>−</sup> *<sup>d</sup>*<sup>2</sup> *dx*<sup>2</sup> *e* on each of the edges *e* ∈ *E* of the graph Γ. The self-adjoint Laplace operator *L*(Γ) has a discrete and non-negative spectrum [12].

A signal inside a graph moves along the edges, and at each vertex *v* ∈ *V* it splits and enters all edges adjacent to *v*. If the signal enters the vertex *v* along the edge *e* and leaves it along the edge *e*, then the ratio of amplitudes of entering and leaving signals is given by the vertex scattering matrix, which depends on the vertex boundary condition. We will consider two types of vertex boundary conditions. The standard boundary conditions are called also Neumann boundary conditions, for which the eigenfunctions are continuous at vertices and the sums of their oriented derivatives at vertices are zero. The vertex scattering matrix corresponding to the Neumann boundary conditions [15] is given by

$${}^{N}\sigma\_{\varepsilon,\varepsilon'}^{(v)} = \frac{2}{d\_v} - \delta\_{\varepsilon,\varepsilon'} \tag{1}$$

where *dv* is the degree of the vertex *v*, i.e., the number of edges incident to the vertex *v*, and *δe*,*e* is the Kronecker delta. The vertices with the Neumann boundary conditions will be denoted as *vN*.

**Citation:** Farooq, O.; Ławniczak, M.; Akhshani, A.; Bauch, S.; Sirko, L. The Generalized Euler Characteristics of the Graphs Split at Vertices. *Entropy* **2022**, *24*, 387. https://doi.org/ 10.3390/e24030387

Academic Editor: Adam Gadomski

Received: 2 February 2022 Accepted: 8 March 2022 Published: 9 March 2022

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For the Dirichlet boundary condition, an eigenfunction at the vertex takes the value zero, which leads to the vertex scattering matrix [15,16]

$$
\sigma^D \sigma^{(v)}\_{\mathfrak{e}, \mathfrak{e}'} = -\delta\_{\mathfrak{e}, \mathfrak{e}'}.\tag{2}
$$

One should point out that the Dirichlet boundary conditions are imposed only at degree one vertices and higher-degree Dirichlet vertices should be treated as separate degree one Dirichlet vertices. The vertices with the Dirichlet boundary conditions will be denoted as *vD*. Different types of the boundary conditions, including the Neumann and Dirichlet ones for higher-dimensional systems such as grains, are comprehensively described in Refs. [17,18].

The total number of vertices |*V*| in a general graph, consisting of both Neumann and Dirichlet boundary conditions, is defined by |*V*| = |*VN*| + |*VD*|, where |*VN*| and |*VD*| denote the number of vertices with Neumann and Dirichlet boundary conditions, respectively.

One of the most important characteristics of metric graphs Γ = (*V*, *E*) with the standard boundary conditions (|*VD*| = 0) is the Euler characteristic

$$\chi = |V| - |E| \, , \tag{3}$$

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 [19–22] 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 the experimental investigation of properties of quantum graphs, we used microwave networks simulating quantum graphs [16,24–29]. The emulation of quantum graphs by microwave networks is possible because of the formal analogy of the one-dimensional Schrödinger equation describing quantum graphs and the telegrapher's equation for microwave networks [24,26]. Microwave networks are the only ones that allow for the experimental simulation of quantum systems with all three types of symmetry within the framework of the random matrix theory (RMT): Gaussian orthogonal ensemble (GOE) systems with preserved time reversal symmetry (TRS) [16,21,24,25,27,30–32], Gaussian unitary ensemble (GUE)—systems with broken TRS [24,28,33–36], and Gaussian symplectic ensemble (GSE)—systems with TRS and half-spin [37]. The other model systems, which are not as versatile as microwave networks, but are often used in simulations of complex quantum systems, are flat microwave billiards [38–54], and exited atoms in strong microwave fields [55–67].

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 |*Vc*| 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). The theoretical results are numerically verified and confirmed experimentally using the spectra of microwave networks simulating quantum graphs.
