Experimental Investigation of the Generalized Euler Characteristic of the Networks Split at Edges

Abstract

We discuss a connection between the generalized Euler characteristic ${\mathcal{E}}_o\left(\right|V_{D_o}\left|\right)$ of the original graph which was split at edges into two separate subgraphs and their generalized Euler characteristics ${\mathcal{E}}_i\left(\right|V_{D_i}\left|\right)$, $i=1,2$, where $|V_{D_o}|$ and $|V_{D_i}|$ are the numbers of vertices with the Dirichlet boundary conditions in the graphs. Applying microwave networks which simulate quantum graphs, we show that the experimental determination of the generalized Euler characteristics ${\mathcal{E}}_o\left(\right|V_{D_o}\left|\right)$ and ${\mathcal{E}}_i\left(\right|V_{D_i}\left|\right)$, $i=1,2$ allows finding the number of edges in which the subnetworks were connected.

1. Introduction

Quantum graphs of connected one-dimensional wires were introduced by Linus Pauling [1] to describe quantum particles in physical networks. At the end of the 1990s, Kottos and Smilansky [2,3] showed that quantum graphs are excellent tools for studying quantum chaos. Later, their various aspects have been the subject of extensive theoretical research [4,5,6,7]. Quantum graphs are also exploited to model a large number of physical systems such as mesoscopic quantum systems [8,9], quantum wires [10], the photon number statistics of coherent light [11] and even quantum interference transistor [12]. However, experimental studies of networks simulating quantum graphs are still very rare [13,14,15].

A metric graph $\mathsf{\Gamma }=(V,E)$ consists of v vertices, $v\in V$, connected by e edges, $e\in E$. The edges e of the graph are segments of the lengths $l_e$ on the real line $\mathbb{R}$. The self-adjoint Laplace operator $L\left(\mathsf{\Gamma }\right)=-\frac{d^2}{d{x}^2}$ is unambiguously determined by the graph. It acts in the Hilbert space of square integrable functions and has discrete and non-negative spectrum [5].

In this article, we will consider two types of vertex boundary conditions: standard and Dirichlet ones. In the case of the standard boundary conditions, called also Kirchhoff or Neumann boundary conditions, the eigenfunctions are continuous at vertices, and the sums of their oriented derivatives at vertices are zero. For the Dirichlet boundary condition, an eigenfunction takes the value zero at the vertex.

The vertex scattering matrices ${}^N{\sigma }_{e,e^{\prime }}^{\left(v\right)}$ and ${}^D{\sigma }_{e,e^{\prime }}^{\left(v\right)}$ corresponding to Neumann and Dirichlet boundary conditions [16] are given by

$${}^N{\sigma }_{e,e^{\prime }}^{\left(v\right)}=\frac{2}{d_v}-{\delta }_{e,e^{\prime }},$$

(1)

and

$$\begin{array}{c}\hfill {}^D{\sigma }_{e,e^{\prime }}^{\left(v\right)}=-{\delta }_{e,e^{\prime }},\end{array}$$

(2)

where $d_v$ is degree of the vertex v, i.e., the number of edges incident to it, and ${\delta }_{e,e^{\prime }}$ is the Kronecker delta.

The Dirichlet boundary conditions can be imposed only at degree one vertices. The total number of vertices in a graph is defined by $\left|V\right|=|V_N|+|V_D|$, where $|V_N|$ and $|V_D|$ designate the numbers of vertices with Neumann and Dirichlet boundary conditions.

An important characteristic of metric graphs $\mathsf{\Gamma }=(V,E)$ with the standard boundary conditions is the Euler characteristic

$$\chi =\left|V\right|-\left|E\right|,$$

(3)

where $\left|V\right|$ and $\left|E\right|$ denote the number of vertices and edges of the graph. Although the Euler characteristic $\chi$ seems to be a purely topological quantity, it has been shown in Refs. [15,17,18,19] that it can be independently defined by the graph and microwave network spectra.

The definition (3) is not valid for the graphs with mixed Neumann and Dirichlet boundary conditions. Therefore, recently, a new spectral invariant, the generalized Euler characteristic $\mathcal{E}$ [15], suitable for graphs and networks with these boundary conditions has been introduced.

From the experimental point of view, it is important that quantum graphs can be simulated using microwave networks [13,16,20]. The simulation of quantum graphs is possible because of the formal analogy of the telegrapher’s equation describing microwave networks [13] and the one-dimensional Schrödinger equation for quantum graphs. Microwave networks permit the simulation of quantum systems described within the framework of the random matrix theory (RMT) by Gaussian orthogonal ensemble (GOE) [13,16,19,21,22] and Gaussian symplectic ensemble (GSE) [14]−systems with preserved time reversal symmetry (TRS), and Gaussian unitary ensemble (GUE)−systems with broken TRS [13,20,23,24]. Thus, any theoretical or numerical results obtained for quantum graphs can be verified experimentally. One should remark that in simulations of the other complex quantum systems, the model systems are often used, such as flat microwave billiards [25,26,27,28,29,30,31,32,33,34,35,36,37,38] and exited atoms in strong microwave fields [39,40,41,42,43,44].

2. The Generalized Euler Characteristic

The Euler characteristic [19] for graphs with the standard boundary conditions at the vertices is given by

$$\chi =2+8{\pi }^2\sum _{\begin{array}{c}{k}_n\in \mathsf{\Sigma }\left(L^{\mathrm{st}}\left(\mathsf{\Gamma }\right)\right)\\ k_n\ne 0\end{array}}\frac{sin(k_n/t)}{(k_n/t)\left({\left(2\pi \right)}^2-{(k_n/t)}^2\right)}{|}_{t\ge t_0},$$

(4)

where $\mathsf{\Sigma }\left(L^{\mathrm{st}}\left(\mathsf{\Gamma }\right)\right)$ stands for the spectrum of the Laplacian $L^{\mathrm{st}}\left(\mathsf{\Gamma }\right)$ with the standard vertex conditions. The number $k_n$ is the square root of the eigenenergy ${\lambda }_n$ and t is a scaling parameter [17,18,19] with $t_0=\frac{1}{2l_{min}}$, where $l_{min}$ is the length of the shortest edge of the graph. The Formula (4), though more complicated, is equivalent to Equation (3). Instead of the number of vertices $\left|V\right|$ and edges $\left|E\right|$, it requires a definite number of the lowest eigenenergies of graphs or networks.

The generalized Euler characteristic for graphs and networks with the mixed boundary conditions (the number of Dirichlet vertices $|V_D|\ne 0$) was defined in Ref. [15]

$${\chi }_G:=\chi -|V_D|=8{\pi }^2\sum _{k_n\in \mathsf{\Sigma }\left(L^{\mathrm{st},\mathrm{D}}\left(\mathsf{\Gamma }\right)\right)}\frac{sin(k_n/t)}{(k_n/t)\left({\left(2\pi \right)}^2-{(k_n/t)}^2\right)}{|}_{t\ge t_0}.$$

(5)

The spectrum of the Laplacian $L^{\mathrm{st},\mathrm{D}}\left(\mathsf{\Gamma }\right)$ with the mixed boundary conditions is denoted by $\mathsf{\Sigma }\left(L^{\mathrm{st},\mathrm{D}}\left(\mathsf{\Gamma }\right)\right)$.

Equations (4) and (5) can be consolidated into a single one for the generalized Euler characteristic

$$\mathcal{E}\left(\right|V_D\left|\right)=2{\delta }_{0,|V_D|}+8{\pi }^2\sum _{\begin{array}{c}{k}_n\in \mathsf{\Sigma }\left(L\left(\mathsf{\Gamma }\right)\right)\\ k_n\ne 0\end{array}}\frac{sin(k_n/t)}{(k_n/t)\left({\left(2\pi \right)}^2-{(k_n/t)}^2\right)}{|}_{t\ge t_0}.$$

(6)

Depending on the boundary conditions, $\mathsf{\Sigma }\left(L\right(\mathsf{\Gamma }\left)\right)$ denotes either the spectrum of the Laplacian $L^{\mathrm{st}}\left(\mathsf{\Gamma }\right)$ or $L^{\mathrm{st},\mathrm{D}}\left(\mathsf{\Gamma }\right)$. For the graphs with the standard boundary conditions, $|V_D|=0$, $\mathcal{E}\left(\right|V_D|=0)=\chi$; however, for the graphs with the mixed boundary conditions, one obtains $|V_D|\ne 0$, $\mathcal{E}\left(\right|V_D|\ne 0)={\chi }_G$, recovering Equations (4) and (5), respectively.

The usability of Equation (6) arises from the fact that $\mathcal{E}\left(\right|V_D\left|\right)$ can be evaluated using only a finite number $K=K_{min}$ of the lowest eigenvalues (resonances) [15,19]

$$K\ge \left|V\right|+2\mathcal{L}t{\left[1-exp\left(\frac{-ϵ\pi }{\mathcal{L}t}\right)\right]}^{-1/2},$$

(7)

where $\mathcal{L}={\sum }_{e\in E}{l}_e$ is the total length of the graph and $ϵ$ denotes the accuracy of determining $\mathcal{E}\left(\right|V_D\left|\right)$ from formula (6). The smallest number of resonances $K_{min}$, at a given accuracy $ϵ$, is obtained if we assign to t its smallest allowed value $t=t_0=\frac{1}{2l_{min}}$. Since the generalized Euler characteristic is an integer, the chosen accuracy should be $ϵ<1/2$. In the calculations of the number of resonances $K_{min}$, $ϵ=1/4$ was assumed.

3. A Graph Split into Two Subgraphs at Edges

In this article, we consider a general situation when an original graph ${\mathsf{\Gamma }}_o\left(\right|V_o|,|E_o|,|V_{D_o}\left|\right)$ is split into two subgraphs ${\mathsf{\Gamma }}_i\left(\right|V_i|,|E_i|,|V_{D_i}\left|\right)$, $i=1,2$, at the common for the subgraphs edges $e_c\in E_c$. A graph or network $\mathsf{\Gamma }\left(\right|V|,|E|,|V_D\left|\right)$ contains $\left|V\right|$ vertices, including $|V_N|$ and $|V_D|$ vertices with Neumann and Dirichlet boundary conditions and $\left|E\right|$ edges. In the partition process, each common edge $e_c$ is split into two new edges belonging to different subgraphs. It is important to point out that the splitting of $|E_c|$ edges is connected with the appearance of $2|E_c|$ new vertices characterized by the degree $d_{v_c}=1$. The splitting of quantum graphs at vertices was considered in Refs. [45,46]. In this general treatment of graphs split at vertices, we assumed that vertices with the degree $d_{v_o}\ge 2$ can be split into two new vertices with $d_{v_o}>d_{v_1}\ge 1$ and $d_{v_2}=d_{v_o}-d_{v_1}\ge 1$. In the case of graphs split at edges, one can also consider a limit case when the cuts are made at the endpoints of the edges. This would be a simplified version of the graphs split at verticies. In this limiting case, each vertex with the degree $d_{v_o}\ge 2$ is split into two new vertices with $d_{v_1}=d_{v_o}-1$ and $d_{v_2}=1$ or $d_{v_1}=1$ and $d_{v_2}=d_{v_o}-1$. This process is theoretically and experimentally described in Refs. [45,46].

Using the definition of the generalized Euler characteristics for the original graph and its subgraphs, one obtains ${\mathcal{E}}_o\left(\right|V_{D_o}\left|\right)=|V_o|-|E_o|-|V_{D_o}|$ and ${\mathcal{E}}_i\left(\right|V_{D_i}\left|\right)=|V_i|-|E_i|-|V_{D_i}|$, $i=1,2$, respectively. Additionally, the relationships between the number of vertices and edges of the graphs are the following: $|V_o|+2|E_c|=|V_1|+|V_2|$ and $|E_o|+|E_c|=|E_1|+|E_2|$. They lead to the relationship between ${\mathcal{E}}_i\left(\right|V_{D_i}\left|\right)$, $i=1,2$ and ${\mathcal{E}}_o\left(\right|V_{D_o}\left|\right)$

$${\mathcal{E}}_1\left(\right|V_{D_1}\left|\right)+{\mathcal{E}}_2\left(\right|V_{D_2}\left|\right)={\mathcal{E}}_o\left(\right|V_{D_o}\left|\right)+|E_c|+|V_{D_o}|-|V_{D_1}|-|V_{D_2}|.$$

(8)

In order to test the Formula (8), let us analyze the graphs presented in Figure 1. The original graph ${\mathsf{\Gamma }}_o\left(\right|V_o|=4,|E_o|=5,|V_{D_o}|=0)={\mathsf{\Gamma }}_o(4,5,0)$ was divided into two subgraphs ${\mathsf{\Gamma }}_1(6,5,0)$ and ${\mathsf{\Gamma }}_2(4,3,3)$. The network ${\mathsf{\Gamma }}_2(4,3,3)$ contains three vertices with the Dirichlet boundary conditions (red dots) and one vertex with the Neumann boundary condition (blue dot). Knowing all parameters of the networks: the numbers of vertices, edges and vertices with the Dirichlet boundary conditions, one can use Equations (3), (5) and (8) to find out that the subgraphs before the disconnection were connected in $|E_c|=3$ common edges. We will show that from the experimental point of view, such complete knowledge about the networks is not always necessary, and $|E_c|$ can be identified just by measuring the spectra of the considered networks.

4. Measurements of the Spectra of Microwave Networks

The generalized Euler characteristic $\mathcal{E}\left(\right|V_D\left|\right)$ defined by Equation (6) was evaluated experimentally by measuring the spectra of microwave networks simulating quantum graphs. Figure 2 shows the experimental set-up which includes an Agilent E8364B vector network analyzer (Agilent, Santa Clara, CA, USA) and HP 85133-60017 flexible microwave cable (Agilent, Santa Clara, CA, USA) that connects the investigated network to VNA. A cable HP 85133-60017 connected to the network is formally equivalent to an infinite lead, which is attached to the quantum graph [15,22]. Using this set-up, the one-port scattering matrices $S_{11}\left(\nu \right)$ of the networks were measured as a function of frequency $\nu$. The modulus of scattering matrix $|S_{11}\left(\nu \right)|$ was used to identify networks’ resonances. In Figure 2, we present also the original microwave network ${\mathsf{\Gamma }}_o(4,5,0)$ which possesses four vertices with the Neumann boundary conditions. As an example of the measured spectra, the spectrum of the network ${\mathsf{\Gamma }}_o(4,5,0)$ is shown in the inset of Figure 2 in the frequency range $\nu =[1.1,2.1]$ GHz.

Microwave networks which simulate quantum graphs consist of microwave junctions and coaxial cables that correspond to the vertices and edges of quantum graphs. The microwave cables (Huber & Suhner, Switzerland) are composed of a center conductor of a radius $r_1=0.05$ cm which is mechanically separated from a cylindrically symmetric conducting shield of an inner radius $r_2=0.15$ cm by the dielectric material (Teflon with the dielectric constant $\epsilon =2.06$). Below the cut-off frequency of the TE${}_{11}$ mode, ${\nu }_{cut}=\frac{c}{\pi (r_1+r_2)\sqrt{\epsilon }}=33$ GHz [47] in the cable propagates only the fundamental TEM mode. The lengths of edges of the simulated quantum graph are equal to the optical lengths of edges of the microwave networks, i.e., $l_{opt}=\sqrt{\epsilon }{l}_{ph}$, where $l_{ph}$ denotes the physical length of the network edges.

5. Results

In this article, we consider two general situations which are possible when the original graph is split at edges into two not connected subgraphs: the case when the original graph and its subgraphs possess only the standard boundary conditions and the case when they are characterized by the mixed boundary conditions, when vertices with the Dirichlet boundary conditions are present. The generalized Euler characteristic $\mathcal{E}\left(\right|V_D\left|\right)$ defined by Equation (6) will be evaluated experimentally by measuring the spectra of the microwave networks.

5.1. Networks with the Standard Boundary Conditions

The original network (graph) ${\mathsf{\Gamma }}_o\left(\right|V_o|,|E_o|,|V_{D_o}\left|\right)$ is split into two disconnected subnetworks (subgraphs) ${\mathsf{\Gamma }}_i\left(\right|V_i|,|E_i|,|V_{D_i}\left|\right)$, $i=1,2$, at the common for the subnetworks (subgraphs) edges $e\in E_c$. All networks possess vertices with the standard boundary conditions. The schemes of the networks ${\mathsf{\Gamma }}_o(4,5,0)$ and its two subnetworks ${\mathsf{\Gamma }}_1(6,5,0)$ and ${\mathsf{\Gamma }}_2(4,3,0)$ are schematically shown in Figure 1. In this case, all Dirichlet boundary conditions (red dots) in ${\mathsf{\Gamma }}_2(4,3,3)$ are changed to the Neumann ones. The standard (Neumann) boundary conditions are denoted in Figure 1 by blue dots.

The experimental identification of microwave networks with the standard boundary conditions is very simple. The existence of their eigenvalue ${\lambda }_0=0$ can be easily detected by measuring the electric conductance $\mathcal{G}$ of the networks. For networks with the standard boundary conditions, we expect $\mathcal{G}=0$ S, while for the networks with at least one Dirichlet boundary condition, $\mathcal{G}=+\infty$ S. The measurements of the electric conductance $\mathcal{G}$ allowed us to find out that the networks ${\mathsf{\Gamma }}_o(4,5,0)$, ${\mathsf{\Gamma }}_1(6,5,0)$, and ${\mathsf{\Gamma }}_2(4,3,0)$ possess no Dirichlet vertices.

The lengths of the graphs ${\mathsf{\Gamma }}_o(4,5,0)$, ${\mathsf{\Gamma }}_1(6,5,0)$, and ${\mathsf{\Gamma }}_2(4,3,0)$ are ${\mathcal{L}}_o=1.802$ m, ${\mathcal{L}}_1=1.115$ m, and ${\mathcal{L}}_2=0.687$ m, respectively. The lengths of their shortest edges are $l_{mi{n}_o}=l_1=0.190$ m, $l_{mi{n}_1}=l_4^1=0.115$ m, and $l_{mi{n}_2}=l_5^2=0.086$ m, giving $K_{mi{n}_o}=28$, $K_{mi{n}_1}=31$, and $K_{mi{n}_2}=23$, respectively, which were estimated using Equation (7). From the experimental point of view, to find the resonances determined by the parameters $K_{mi{n}_o}$, $K_{mi{n}_1}$, and $K_{mi{n}_2}$, it was necessary to measure the spectra of the microwave networks ${\mathsf{\Gamma }}_o(4,5,0)$, ${\mathsf{\Gamma }}_1(6,5,0)$, and ${\mathsf{\Gamma }}_2(4,3,0)$ in the frequency ranges $[0.010,2.35]$ GHz, $[0.010,4.16]$ GHz, and $[0.010,5.00]$ GHz, respectively. Then, using the found resonance positions, the generalized Euler characteristics ${\mathcal{E}}_o\left(\right|V_{D_o}\left|\right)$, ${\mathcal{E}}_1\left(\right|V_{D_1}\left|\right)$, and ${\mathcal{E}}_2\left(\right|V_{D_2}\left|\right)$ were evaluated using Equation (6). In Figure 3a–c, we show the generalized Euler characteristics ${\mathcal{E}}_o\left(\right|V_{D_o}|=0)$ (blue full line), ${\mathcal{E}}_1\left(\right|V_{D_1}|=0)$ (blue dashed line), and ${\mathcal{E}}_2\left(\right|V_{D_2}|=0)$ (blue dotted line) evaluated experimentally as a function of the parameter t. The plateaux of the generalized Euler characteristics start close to the points $t_{0_o}=2.63$ m${}^{-1}$, $t_{0_1}=4.33$ m${}^{-1}$, and $t_{0_2}=5.85$ m${}^{-1}$, respectively. The values of $t_{0_i}$, $i=o,1,2$ are marked in Figure 3 by vertical arrows. The values of the generalized Euler characteristics are found to be ${\mathcal{E}}_o\left(\right|V_{D_o}|=0)=-1$, ${\mathcal{E}}_1\left(\right|V_{D_1}|=0)=1$, and ${\mathcal{E}}_2\left(\right|V_{D_2}|=0)=1$, respectively. Using Equation (8), one can find out that $|E_c|=3$, showing that before splitting, the two subnetworks were connected at three edges. It is important to underline that from the experimental point of view, in the case of the standard boundary conditions, the number of common edges $|E_c|$ can be identified just by measuring the eigenfrequencies of the networks without any need of seeing the networks or knowing their topologies.

5.2. Networks with the Mixed Boundary Conditions

The same networks were used to investigate the splitting at edges of the original network ${\mathsf{\Gamma }}_o(4,5,0)$ into two separated subnetworks ${\mathsf{\Gamma }}_1(6,5,0)$ and ${\mathsf{\Gamma }}_2(4,3,3)$. The subnetwork ${\mathsf{\Gamma }}_2(4,3,3)$ possesses three vertices with the Dirichlet boundary conditions (see Figure 1). The other parameters of the networks are the same as for the networks with the standard boundary conditions. Although Equation (7) yields the same number of resonances for the networks with the mixed boundary conditions and for the ones with the Neumann boundary conditions, the frequency ranges where they are located are slightly different. For example, for the network ${\mathsf{\Gamma }}_2(4,3,3)$, it is $[0.010,5.02]$ GHz compared to the same network ${\mathsf{\Gamma }}_2(4,3,0)$ with the Neumann boundary conditions for which the frequency range is $[0.010,5.00]$ GHz. The measurements of the electric conductance of the networks yielded that the networks ${\mathsf{\Gamma }}_o(4,5,0)$ and ${\mathsf{\Gamma }}_1(6,5,0)$ possess only vertices with the Neumann boundary conditions, while the network $\mathsf{\Gamma }(4,3,3)$ possesses at least one Dirichlet vertex.

In Figure 4a–c, we show the generalized Euler characteristics ${\mathcal{E}}_o\left(\right|V_{D_o}|=0)$ (blue full line), ${\mathcal{E}}_1\left(\right|V_{D_1}|=0)$ (blue dashed line), and ${\mathcal{E}}_2\left(\right|V_{D_2}|=3)$ (red dotted line) evaluated experimentally as a function of the parameter t. In addition, in all three cases, the plateaux of the generalized Euler characteristics starts close to the points $t_{0_o}$, $t_{0_1}$, and $t_{0_2}$ defined by the theory. We found out that the values of the generalized Euler characteristics are the following ${\mathcal{E}}_o\left(\right|V_{D_o}|=0)=-1$, ${\mathcal{E}}_1\left(\right|V_{D_1}|=0)=1$, and ${\mathcal{E}}_2\left(\right|V_{D_2}|=3)=-2$.

Using Equation (8) with the condition $|V_{D_2}|=3$, we again find out that $|E_c|=3$. It is important to point out that for the networks with the mixed boundary conditions, the knowledge of their topologies does not allow one to find their generalized Euler characteristics. Additionally to that, we have to know the number of their vertices with the Dirichlet boundary conditions. However, the measurements of the electric conductance of the networks can warn us that we deal with the networks which possess at least one Dirichlet vertex.

6. Summary

We put to the experimental test a formula which links the generalized Euler characteristic ${\mathcal{E}}_o\left(\right|V_{D_o}\left|\right)$ of the original network (graph) which was split at edges into two not connected subnetworks (subgraphs) with their generalized Euler characteristics ${\mathcal{E}}_i\left(\right|V_{D_i}\left|\right)$, $i=1,2$. We demonstrated that the experimental evaluation of the generalized Euler characteristics ${\mathcal{E}}_o\left(\right|V_{D_o}\left|\right)$ and ${\mathcal{E}}_i\left(\right|V_{D_i}\left|\right)$, $i=1,2$, also in the case of the mixed boundary conditions, allows one to determine the number $|E_c|$ of the edges where the two subnetworks were initially connected. This method is especially useful for the networks with the standard boundary conditions because it allows one to find $|E_c|$ without knowing their topologies.