Role of the Boundary Conditions in the Graphs Split at Vertices

Abstract

We discuss a role of the boundary conditions in the graphs split at vertices. In the analysis we take advantage of the properties of a new spectral invariant for quantum graphs and networks: the generalized Euler characteristics $$\mathcal E_i(|V_{D_i}|) $$, where $$|V:{D_i}|$$ denotes the number of vertices with the Dirichlet boundary conditions in the ith graph. The theoretical results are verified numerically.