*Article* **The Generalized Euler Characteristics of the Graphs Split at Vertices**

**Omer Farooq \*, Michał Ławniczak \*, Afshin Akhshani, Szymon Bauch and Leszek Sirko \***

Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warsaw, Poland; akhshani@ifpan.edu.pl (A.A.); bauch@ifpan.edu.pl (S.B.)

**\*** Correspondence: omer.farooq@ifpan.edu.pl (O.F.); lawni@ifpan.edu.pl (M.Ł.); sirko@ifpan.edu.pl (L.S.)

**Abstract:** We show that there is a relationship between the generalized Euler characteristic E*o*(|*VDo* |) of the original graph that was split at vertices into two disconnected subgraphs *i* = 1, 2 and their generalized Euler characteristics E*i*(|*VDi* |). Here, |*VDo* | and |*VDi* | denote the numbers of vertices with the Dirichlet boundary conditions in the graphs. The theoretical results are experimentally verified using microwave networks that simulate quantum graphs. We demonstrate that the evaluation of the generalized Euler characteristics E*o*(|*VDo* |) and E*i*(|*VDi* |) allow us to determine the number of vertices where the two subgraphs were initially connected.

**Keywords:** quantum graphs; microwave networks; Euler characteristic; Neumann and Dirichlet boundary conditions
