*3.1. Networks with the Standard Boundary Conditions*

Here, we will consider the original network Γ*o*(|*Vo*|, |*Eo*|, |*VDo* |), which is split into two disconnected subnetworks Γ*i*(|*Vi*|, |*Ei*|, |*VDi* |), *i* = 1, 2, at the common for the subnetworks vertices *v* ∈ *Vc*. All networks are characterized by the standard (Neumann) boundary conditions. The experimental realizations of the networks Γ*o*(6, 9, 0) and its two subnetworks Γ1(4, 6, 0) and Γ2(4, 3, 0) are schematically shown in Figures 1 and 2. In this case, all networks possess only standard (Neumann) boundary conditions, denoted with the capital letter *N*.

The total optical lengths of the networks Γ*o*(6, 9, 0), Γ1(4, 6, 0), and Γ2(4, 3, 0) are L*<sup>o</sup>* = 2.579 m, L<sup>1</sup> = 1.675 m, and L<sup>2</sup> = 0.940 m, respectively. The lengths of their shortest edges are *lmino* = *l*<sup>6</sup> = 0.221 m, *lmin*<sup>1</sup> = *l*<sup>6</sup> = 0.221 m, and *lmin*<sup>2</sup> = *l*<sup>9</sup> = 0.270 m, giving *Kmino* = 38, *Kmin*<sup>1</sup> = 23, and *Kmin*<sup>2</sup> = 8, respectively, which were estimated using Equation (7). Experimentally, in order to find the minimum number of resonances determined by the parameters *Kmino* , *Kmin*<sup>1</sup> , and *Kmin*<sup>2</sup> , it was necessary to measure the spectra of the microwave networks Γ*o*(6, 9, 0), Γ1(4, 6, 0), and Γ2(4, 3, 0) in the frequency ranges

[0.010, 2.347] GHz, [0.010, 2.234] GHz, and [0.010, 1.271] GHz, respectively. Taking into account the above parameters, the generalized Euler characteristics E*o*(|*VDo* |), E1(|*VD*<sup>1</sup> |), and E2(|*VD*<sup>2</sup> |) were calculated using Equation (6).

In Figure 3a–c, we show the generalized Euler characteristics E*o*(|*VDo* | = 0), E1(|*VD*<sup>1</sup> | = 0), and E2(|*VD*<sup>2</sup> | = 0) (red dotted lines), evaluated experimentally as a function of the parameter *t*. The numerically found generalized Euler characteristics are marked with blue full lines. 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>* = 2.26 m−1, *t*<sup>01</sup> = 2.26 m<sup>−</sup>1, and *t*<sup>02</sup> = 1.85 m−<sup>1</sup> defined by the theory (see the discussion below Equation (7)). The values of the generalized Euler characteristics are found to be E*o*(|*VDo* | = 0) = −3, E1(|*VD*<sup>1</sup> | = 0) = −2, and E2(|*VD*<sup>2</sup> | = 0) = 1, respectively. Using Equation (8), it is easy to find that |*Vc*| = 2. It means that, before splitting, the two subgraphs were connected at the two vertices. It is important to point out that the above information was obtained without knowing anything about the topologies of the networks.
