Experimental Investigation of the Generalized Euler Characteristic of the Networks Split at Edges

Abstract

We discuss a connection between the generalized Euler characteristic Eo(|VDo|) of the original graph which was split at edges into two separate subgraphs and their generalized Euler characteristics Ei(|VDi|), i=1,2, where |VDo| and |VDi| are the numbers of vertices with the Dirichlet boundary conditions in the graphs. Applying microwave networks which simulate quantum graphs, we show that the experimental determination of the generalized Euler characteristics Eo(|VDo|) and Ei(|VDi|), i=1,2 allows finding the number of edges in which the subnetworks were connected.

Keywords

• microwave networks
• quantum graphs
• Euler characteristic
• Neumann and Dirichlet boundary conditions