Investigation of the generalized Euler characteristic of graphs and microwave networks split at edges and vertices

We analyze the situation when the original graph is split at edges and vertices into two disconnected subgraphs. We show that there is a relationship between the generalized Euler characteristic o(∣VDo∣) of the original graph and the generalized Euler characteristics i(∣VDi∣) , i = 1,2, of two disconnected subgraphs, where ∣VDo∣ and ∣VDi∣ , i = 1,2, are the numbers of vertices with the Dirichlet boundary conditions in the graphs. Theoretical predictions are verified experimentally using microwave networks which simulate quantum graphs.

A metric graph Γ = (V, E) consists of v vertices, v ä V, which are connected by e edges of the lengths l e , e ä E. A metric graph becomes a quantum one after being equipped with the self-adjoint Laplace operator L d dx 2 2 ( ) G =acting in the Hilbert space of square-integrable functions, which has a discrete and non-negative spectrum [6].
In this article two types of vertex boundary conditions will be considered: Neumann and Dirichlet ones. In the case of the Neumann boundary conditions, the eigenfunctions are continuous at vertices and the sums of their oriented derivatives at vertices are zero. For the Dirichlet boundary conditions eigenfunctions take the value zero at the vertices. conditions [16,17] are defined respectively by where d v is the degree of the vertex v, i.e. the number of edges connected to the vertex v, and e e , d ¢ is the Kronecker delta.
It is important to point out that the Dirichlet boundary conditions can be imposed only at the vertices of the degree one. The total number of vertices |V| in a graph with Neumann and Dirichlet boundary conditions is denoted by |V| = |V N | + |V D |, where |V N | and |V D | stand for the numbers of vertices with Neumann and Dirichlet boundary conditions, respectively.
The Euler characteristic χ is one of the most important characteristic of metric graphs Γ = (V, E) with the Neumann boundary conditions Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
where |V| and |E| denote the number of vertices and edges of the graph. Until recently the Euler characteristic χ has been considered to be a pure topological quantity. However, it has been shown in [18][19][20] and [21] that it can be independently defined and evaluated using theh spectra of quantum graphs or microwave networks. The definition of the Euler characteristic (3) is not valid for the graphs with mixed Neumann and Dirichlet boundary conditions. To rectify this situation a new spectral invariant, the generalized Euler characteristic  [21,22], suitable for graphs and networks with the mixed boundary conditions has been recently introduced.

The generalized Euler characteristic
The Euler characteristic χ for graphs with the Neumann boundary conditions at the vertices [20] is defined as where Σ(L N (Γ)) denotes the spectrum of the Laplacian L N (Γ) with the Neumann boundary conditions. The wavenumbers k n are the square roots of the eigenenergies λ n and t is a scaling parameter [18][19][20] with t l 0 , where l min is the length of the shortest edge of the graph. The formula (4) is equivalent to equation (3) but instead of using topological information about graphs, the number of vertices |V| and edges |E|, it requires a certain number of the lowest eigenenergies (resonances) of graphs or networks.
The generalized Euler characteristic χ G for graphs and networks with mixed boundary conditions, the Neumann and Dirichlet ones (the number of Dirichlet vertices |V D | ≠ 0), was defined in [21] The spectrum of the Laplacian L N,D (Γ) with the mixed boundary conditions is denoted by Σ(L N,D (Γ)).
The equations (4) and (5) can be written in a single form Depending on the boundary conditions Σ(L(Γ)) denotes either the spectrum of the Laplacian L N (Γ) or L N,D (Γ).
For the graphs with the Neumann boundary conditions, The usefulness of equation (6) stems from the fact that in most cases V D (| |) can be evaluated using only a small number K K min = of the lowest eigenvalues of the considered graphs [20,21,64,65] where l e E e  = å Î is the length of the graph and ò is the accuracy of determining the Euler characteristic by formula (6). The smallest number of resonances K min , for a given accuracy ò, is obtained if we assign to t its smallest allowed value t l 0 Since the generalized Euler characteristic is an integer, the chosen accuracy should be ò < 1/2. In the calculations presented in this article the number of resonances K min was evaluated assuming that ò = 1/4.

A graph split at edges and vertices into two subgraphs
In this article we consider a general situation when an original graph , i = 1,2, at the common for the subgraphs edges e c ä E c and vertices v c ä V c . A graph or network Γ(|V|, |E|, |V D |) contains |V| vertices, including |V N | = |V| − |V D | and |V D | vertices with Neumann and Dirichlet boundary conditions and |E| edges. In the partition process the common edges e c and vertices v c are split into new edges and vertices belonging to different subgraphs. The splitting of |E c | edges is connected with the appearance of |E c | new edges and 2|E c | new vertices characterized by the degree d 1 v c = . In the process of splitting of |V c | vertices, |V c | new vertices will appear with the degree . Simpler situations of quantum graphs split only at vertices or at edges were considered in [66,67] and [68], respectively.
By definition the generalized Euler characteristics of the original graph and its subgraphs are Whereas the relationships between the number of vertices and edges of the graphs are the following: The formula (10) will be experimentally tested using microwave networks simulating quantum graphs.
In figure 1(a) we show the scheme of the original graph which was divided at two edges |E c | = 2 and two vertices |V c | = 2 into two subgraphs Γ 1 (6, 8, 0) and Γ 2 (5, 4, 0). The analyzed graphs contain only vertices with the Neumann boundary conditions which are marked with blue dots and capital letters N. Since we know all topological parameters of the networks we can use equations (3) and (10) to find out that the subgraphs before disconnection were connected in |E c | + |V c | = 4 common points, such as vertices, edges or both. From equation (9) we can additionally find out that |E c | = 2, thus obtaining the number of common vertices to be |V c | = 2.
In figure 1(b) we show more complicated situation when the original graph Γ o (5, 10, 0) was divided at two edges |E c | = 2 and two vertices |V c | = 2 into two subgraphs Γ 1 (6, 8, 0) and Γ 2 (5,4,4). In this case the graph Γ 2 (5, 4, 4) contains four vertices with the Dirichlet boundary conditions (red dots and capital letters D) and one vertex with the Neumann boundary condition (blue dot and the capital letter N). Applying equations (3), (5), and (10) we again find out that the subgraphs before the disconnection were connected in |E c | + |V c | = 4 common points. The same as above analysis leads to the identification of two common edges |E c | = 2 and two common vertices |V c | = 2 where the graphs were connected before splitting.
Later in this article we will show that from the experimental point of view the complete knowledge about the topology of the graphs is not always necessary and |E c | + |V c | can be identified just from the spectra of the considered networks simulating quantum graphs.

Microwave networks and their spectra
Microwave networks simulating quantum graphs consist of coaxial cables and junctions that correspond to the edges and vertices of quantum graphs. The microwave cables are constructed from a central conductor wire of a radius r 1 = 0.05 cm which is electrically separated from a cylindrically symmetric conducting shield of an inner radius r 2 = 0.15 cm by Teflon, dielectric material with the dielectric constant ε = 2.06. Below the cut-off frequency 33 GHz , where l ph is the physical length of a network edge. In order to evaluate experimentally the generalized Euler characteristic V D (| |) defined by equation (6) we measured the spectra of microwave networks simulating quantum graphs presented in figures 1(a)-(b). Figure 2 shows the experimental set-up applied for this purpose. It consists of Agilent E8364B vector network analyzer (VNA) and a flexible microwave cable HP 85133-60016 that connects the network to VNA. A flexible cable HP 85133-60016 connected to the network is equivalent to an infinite lead which is attached to the quantum graph [21,31]. Utilizing this set-up one-port scattering matrices S 11 (ν) of the networks were measured as a function of frequency ν. The modulus of |S 11 (ν)| was used to find and identify resonances of the networks. In

Results
We will discuss two general situations which are possible when the original network is split at edges and vertices into two subnetworks: the case when the original network and its subnetworks possess only the Neumann boundary conditions and the case when some of the networks are characterized by the mixed boundary conditions. The generalized Euler characteristics V D (| |) of the networks defined by equation (6) will be evaluated experimentally from the spectra of the investigated networks.

Networks with the Neumann boundary conditions
In this case the original network , i = 1,2, at the common for the subnetworks edges e ä E c and vertices e ä V c . All networks are characterized by the Neumann boundary conditions. The schemes of the networks Γ o (5, 10, 0) and its two subnetworks Γ 1 (6, 8, 0) and Γ 2 (5, 4, 0) are schematically shown in figure 1(a). The Neumann boundary conditions are denoted in figure 1(a)    (| | ) = = , respectively. Then using equation (10) we can find out that |E c | + |V c | = 4. It means that before the splitting the two subnetworks were connected at 4 points: edges, vertices or both. One should point out that in the case of the Neumann boundary conditions the number |E c | + |V c | can be identified just on the basis of the networks' spectra without any need to know their topologies.

Networks with the mixed boundary conditions
We applied the same networks to investigate the splitting at edges and vertices of the original network Γ o (5, 10, 0) into two separated subnetworks Γ 1 (6, 8, 0) and Γ 2 (5,4,4). In this case the subnetwork Γ 2 (5,4,4) possesses four vertices with the Dirichlet boundary conditions (see figure 1(b)). All other parameters of the networks are the same as in the case of the networks with the Neumann boundary conditions which were discussed above. For the networks with the mixed boundary conditions equation (7)  (| | ) = (red dotted line) evaluated experimentally as a function of the parameter t. As expected, the plateaux at the generalized Euler characteristics start close to the points t 0 o , t 0 1 , and t 0 2 defined by the theory. We found out that the experimental values of the generalized Euler characteristics are the following: Applying equation (10) with the condition V 4 D 2 | | = we again find out that |E c | + |V c | = 4. However, one should point out that for the networks with the mixed boundary conditions the correct number |E c | + |V c | can only be identified if we know the number of their vertices with the Dirichlet boundary conditions. From this point of view the measurements of the electric conductance of the networks are crucial. They can warn us that we are dealing with the networks that possess at least one vertex with the Dirichlet boundary condition.  (| |), i = 1, 2, also in the case of the mixed boundary conditions allow to determine the sum of the edges and vertices |E c | + |V c | where the two subgraphs (subnetworks) were initially connected. The presented method appeared to be especially useful for the graphs (networks) with the Neumann boundary conditions because it allows to find |E c | + |V c | without knowing graphs (networks) topologies. In this case if we additionally know either |E c | or |V c |, the generalized Euler characteristic defined by equation (6) allows us to find accordingly the exact number of vertices |V c | or edges |E c | where the original graph (network) was split.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.