The Generalized Euler Characteristics of the Graphs Split at Vertices

We show that there is a relationship between the generalized Euler characteristic [Formula: see text] of the original graph that was split at vertices into two disconnected subgraphs [Formula: see text] and their generalized Euler characteristics [Formula: see text]. Here, [Formula: see text] and [Formula: see text] 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 [Formula: see text] and [Formula: see text] allow us to determine the number of vertices where the two subgraphs were initially connected.