AbstractThe Euler characteristic i.e., the difference between the number of vertices |V| and edges |E| is the most important topological characteristic of a graph. However, to describe spectral properties of differential equations with mixed Dirichlet and Neumann vertex conditions it is necessary to introduce a new spectral invariant, the generalized Euler characteristic $$\chi _G:= |V|-|V_D|-|E|$$
χ
G
:
=
|
V
|
-
|
V
D
|
-
|
E
|
, with $$|V_D|$$
|
V
D
|
denoting the number of Dirichlet vertices. We demonstrate theoretically and experimentally that the generalized Euler characteristic $$\chi _G$$
χ
G
of quantum graphs and microwave networks can be determined from small sets of lowest eigenfrequencies. If the topology of the graph is known, the generalized Euler characteristic $$\chi _G$$
χ
G
can be used to determine the number of Dirichlet vertices. That makes the generalized Euler characteristic $$\chi _G$$
χ
G
a new powerful tool for studying of physical systems modeled by differential equations on metric graphs including isoscattering and neural networks where both Neumann and Dirichlet boundary conditions occur.
A new spectral invariant for quantum graphs
