Hearing Euler characteristic of graphs

Abstract

The Euler characteristic χ = |V | − |E| and the total length L are the most important topological and geometrical characteristics of a metric graph. Here, |V | and |E| denote the number of vertices and edges of a graph. The Euler characteristic determines the number β of independent cycles in a graph while the total length determines the asymptotic behavior of the energy eigenvalues via the Weyl's law. We show theoretically and confirm experimentally that the Euler characteristic can be determined (heard) from a finite sequence of the lowest eigenenergies λ 1 ,. .. , λ N of a simple quantum graph, without any need to inspect the system visually. In the experiment quantum graphs are simulated by microwave networks. We demonstrate that the sequence of the lowest resonances of microwave networks with β ≤ 3 can be directly used in determining whether a network is planar, i.e., can be embedded in the plane. Moreover, we show that the measured Euler characteristic χ can be used as a sensitive revealer of the fully connected graphs.

A new spectral invariant for quantum graphs

“…Here, we will generalize the results obtained in Ref. 18 to the case of mixed standard (Neumann) and Dirichlet vertex conditions. We assume that Dirichlet conditions are imposed only at degree one vertices of pendent edges since higher degree Dirichlet vertices should be treated as separate degree one Dirichlet vertices.…”

“…This case was comprehensively treated in Ref. 18 . Here, we will generalize the results obtained in Ref.…”

“…For graphs with standard vertex conditions at all vertices the following formula for the Euler characteristic was proven 18 : where denotes the spectrum of the Laplacian with standard vertex conditions, taken in the square root scale, i.e., the numbers are the square roots of the eigenenergies and , where is the length of the shortest edge of the graph. If the summation sequence in Eq.…”

“…In this article we generalize the notion of the Euler characteristic 18 to graphs possessing vertices with both Neumann and Dirichlet boundary conditions. We show that for such graphs it is possible to determine the generalized Euler characteristic from small sets of their lowest eigenvalues .…”

“…The microwave networks are extremely useful in studying quantum and wave chaos. Uniquely they allow for the experimental realization of systems described by the main three symmetry classes in random-matrix theory (RMT): systems with preserved time reversal symmetry (TRS) represented by Gaussian orthogonal ensemble (GOE) 18 – 21 , 23 , 25 , 26 ; systems with preserved TRS and half-spin represented by Gaussian symplectic ensemble (GSE) 27 , 28 ; systems without TRS represented by Gaussian unitary ensemble (GUE) 19 , 24 , 29 – 33 . The chiral orthogonal, unitary, and symplectic ensembles 34 have been recently realized using microwave networks.…”