Investigation of the generalized Euler characteristic of graphs and microwave networks split at edges and vertices

Abstract

Abstract We analyze the situation when the original graph is split at edges and vertices into two disconnected subgraphs. We show that there is a relationship between the generalized Euler characteristic  o ( ∣ V D o ∣ ) of the original graph and the generalized Euler characteristics  i ( ∣ V D i ∣ ) , i = 1,2, of two disconnected subgraphs, where ∣ V D o ∣ and ∣ V D i ∣ , i = 1,2, are the numbers of vertices with the Dirichlet boundary conditions in the graphs. Theoretical predictions are verified experimentally using microwave networks which simulate quantum graphs.