**4. Summary**

We analyzed a relationship between the generalized Euler characteristic E*o*(|*VDo* |) of the original graph (network), which was split into two disconnected subgraphs (subnetworks) *i* = 1, 2, and their generalized Euler characteristics E*i*(|*VDi* |). We showed that the evaluation of the generalized Euler characteristics E*o*(|*VDo* |) and E*i*(|*VDi* |) allows us

to determine the number |*Vc*| of common vertices where the two subgraphs were initially connected. The theoretical results were numerically verified and confirmed experimentally using microwave networks with the standard and mixed boundary conditions. The application of the generalized Euler characteristics defined by Equation (6) requires the measurement of the spectra of the networks but in return allows us to find |*Vc*| without knowing their topologies. Therefore, it might be possible to apply the properties of the splitting networks discussed in this article in some more practical applications, such as the diagnostics of electronic or microwave networks. One should underline that the first practical test of such diagnostics where the properties of splitting networks and the generalized Euler characteristic were applied was presented in this article. For this purpose, we used real-world systems, such as microwave networks. They are open and dissipative systems, which are completely different from the ideal dissipationless graphs considered in their mathematical studies. In spite of this, even for more complex networks possessing the mixed boundary conditions, we were able to find experimentally the number of common vertices |*Vc*| where the two separated subnetworks were connected before their splitting.

**Author Contributions:** O.F. performed the experiment. M.Ł. performed numerical calculations. O.F., M.Ł., A.A. and S.B. performed the data analysis. L.S. provided the theoretical interpretation. L.S. wrote the manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research study was funded by the National Science Centre, Poland, Grant No. 2018/30/Q/ ST2/00324.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data that support the results presented in this paper and other findings of this study are available from the corresponding authors upon reasonable request.

**Conflicts of Interest:** The authors declare no conflict of interest.
