*3.2. Networks with the Mixed Boundary Conditions*

We used the same physical networks to investigate the split of the original network Γ*o*(6, 9, 1) possessing the mixed boundary conditions into two separated subnetworks Γ1(4, 6, 0) and Γ2(4, 3, 1). The network Γ*o*(6, 9, 1) and the subnetwork Γ2(4, 3, 1) possess a single Dirichlet boundary condition. Figure 1 shows the schemes of the networks. The Dirichlet boundary conditions are denoted by the capital letter *D*. All other parameters of the networks, such as the total lengths and the shortest edges, are the same as in the case of the networks with the standard boundary conditions, which were discussed above. However, for the networks with the mixed boundary conditions, one requires the same number of resonances as, in the case of the networks with the Neumann boundary conditions, the frequency ranges where they can be identified are different. For example, for the networks Γ*o*(6, 9, 1) and Γ2(4, 3, 1), they are [0.010, 2.500] GHz and [0.010, 1.131] GHz, respectively.

In Figure 4a–c, we show the generalized Euler characteristics E*o*(|*VDo* | = 1), E1(|*VD*<sup>1</sup> | = 0), and E2(|*VD*<sup>2</sup> | = 1) (red dotted lines), evaluated experimentally as a function of the parameter *t*. The generalized Euler characteristics that were found numerically are marked with blue full lines. Moreover, here, in all three cases, for both experimental and theoretical results, the plateaus at the generalized Euler characteristics start close to the points *t*0*<sup>o</sup>* , *t*01 , and *t*<sup>02</sup> defined by the theory. The values of the generalized Euler characteristics are found to be E*o*(|*VDo* | = 1) = −4, E1(|*VD*<sup>1</sup> | = 0) = −2, and E2(|*VD*<sup>2</sup> | = 1) = 0, respectively. In addition, in this case, using Equation (8), we found that |*Vc*| = 2. One should remark that in the case of the mixed boundary conditions, the knowledge of the topologies of the experimental networks does not allow us to find their generalized Euler characteristics. We also have to know the number of their Dirichlet boundary conditions. Therefore, the measurements of the spectra of the networks and using Equation (6) are mandatory.

**Figure 3.** Generalized Euler characteristics evaluated for the networks with the standard boundary conditions as a function of the parameter *t*. Panels (**a**–**c**) show the generalized Euler characteristics E*o*(|*VDo* | = 0), E1(|*VD*<sup>1</sup> | = 0), and E2(|*VD*<sup>2</sup> | = 0) of the networks Γ*o*(6, 9, 0), Γ1(4, 6, 0), and Γ2(4, 3, 0), respectively. The experimental and numerical results are marked with red dotted and blue full lines, respectively. In all three cases, the plateaus at the generalized Euler characteristics start close to the points *t*0*<sup>o</sup>* = 2.26 m−1, *t*<sup>01</sup> = 2.26 m−1, and *t*<sup>02</sup> = 1.85 m−1, respectively, defined by the theory (see the discussion below Equation (7)). The black broken lines show the limits of the expected errors E*q*(|*VDq* |) ± 1/4, where *q* = *o*, 1, and 2.

**Figure 4.** Generalized Euler characteristics evaluated for the networks with the mixed boundary conditions as a function of the parameter *t*. Panels (**a**–**c**) show the generalized Euler characteristics E*o*(|*VDo* | = 1), E1(|*VD*<sup>1</sup> | = 0), and E2(|*VD*<sup>2</sup> | = 1) of the networks Γ*o*(6, 9, 1), Γ1(4, 6, 0), and Γ2(4, 3, 1), respectively. The experimental and numerical results are marked with red dotted and blue full lines, respectively. Moreover, here, in all three cases, the plateaus at the generalized Euler characteristics start close to the points *t*0*<sup>o</sup>* = 2.26 m−1, *t*<sup>01</sup> = 2.26 m−1, and *t*<sup>02</sup> = 1.85 m−1, respectively, defined by the theory. The black broken lines show the limits of the expected errors E*q*(|*VDq* |) ± 1/4, where *q* = *o*, 1, and 2.
