**3. Measurements of the Spectra of Microwave Networks**

In order to evaluate the generalized Euler characteristic E(|*VD*|) defined by Equation (6), we measured the spectra of microwave networks simulating quantum graphs. 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*. The modulus of |*S*11(*ν*)| was used to identify the network's resonances. In Figure 2, we also show the original microwave network Γ*o*(6, 9, 1), which possesses a single vertex with the Dirichlet boundary condition (*VDo* = 1), marked by the red capital letter *D*. The measured spectrum of the network Γ*o*(6, 9, 1) is shown in the inset of Figure 2 in the frequency range *ν* = [0.01, 1] GHz. In order to reconfirm our experimental results, the spectra of the

quantum graphs simulated by the microwave networks were also calculated numerically using the pseudo-orbits method developed in Ref. [31].

**Figure 2.** The experimental set-up. It contains an Agilent E8364B vector network analyzer (VNA) and HP 85133-60016 flexible microwave cable that connects the VNA to the measured network. The original microwave network Γ*o*(6, 9, 1) possesses a single vertex with the Dirichlet boundary condition, which is marked by the red capital letter *D*. The measured spectrum of the network Γ*o*(6, 9, 1) is shown in the inset in the frequency range *ν* = [0.01, 1] GHz.

In the construction of microwave networks simulating quantum graphs, we used microwave coaxial cables and junctions that corresponded to the edges and vertices of the quantum graphs. The microwave cables consisted of an outer conductor with an inner radius *r*<sup>2</sup> = 0.15 cm and an inner conductor of radius *r*<sup>1</sup> = 0.05 cm, which was surrounded by the dielectric material (Teflon). The fundamental TEM mode propagates in such cables below the cut-off frequency of the TE11 mode *νcut* = *<sup>c</sup> π*(*r*1+*r*2) <sup>√</sup>*<sup>ε</sup>* = 33 GHz [70,71], where the dielectric constant of Teflon *ε* = 2.06. It is important to point out that the lengths of edges of the simulated quantum graph have to be compared to the optical lengths of the edges of the microwave networks, i.e., *lopt* <sup>=</sup> <sup>√</sup>*εlph*, where *lph* is the physical length of the network edges.

In this paper, we discuss two general situations that are possible when the original network (graph) is split into two subnetworks (subgraphs): the case when the original network and its subnetworks have only the standard boundary conditions and the case when they are characterized by the mixed boundary conditions, when the Dirichlet boundary conditions are present.
