ACTA PHYSICA POLONICA A No. 6 Vol. 140 (2021) Proceedings of the 10th Workshop on Quantum Chaos and Localisation Phenomena (CHAOS 21) Some Applications of Generalized Euler Characteristic of Quantum Graphs and Microwave Networks S. Bauch a,* , M. Lawniczak a , J. Wrochna a , P. Kurasov b and L. Sirko a a Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warszawa, Poland b Department of Mathematics, Stockholm University, S-106 91 Stockholm, Sweden Doi: 10.12693/APhysPolA.140.525 ∗ e-mail: bauch@ifpan.edu.pl In this article we continue to explore the possibilities offered by our discovery that one of the main graph and network characteristic, the generalized Euler characteristic χG, can be determined from a graph/network spectrum. We show that using the generalized Euler characteristic χG the number of vertices with Dirichlet |VD| boundary conditions of a family of graphs/networks created on the basis of the standard quantum graphs or microwave networks can be easily identified. We also present a new application of the generalized Euler characteristic for checking the completeness of graphs/networks spectra in the low energy range. topics: quantum graphs, Euler characteristic, boundary conditions, microwave networks 1. Introduction The concept of a graph was introduced by Leon- hard Euler in XVIII century [1]. Exactly two hun- dred years later, Linus Pauling published an ar- ticle [2] in which he used graphs to describe the motion of quantum particles in a physical network. This approach is called quantum graph model and is widely used to investigate many physical sys- tems, e.g. mesoscopic quantum system [3, 4], quan- tum wires [5] and optical wave guides [6]. Richard P. Feynman [7] applied diagrams (graphs) as pic- torial representation of the mathematical expres- sions describing the behavior and interaction of sub- atomic particles. Due to the extremely wide range of applications the theory of quantum graphs has been the subject of extensive research so far [8–14]. In particular, Kottos and Smilansky [9] showed that quantum graphs can be used to study quantum chaos, i.e., the phenomena found in quantum sys- tems that are chaotic at the classical limit. The metric graph Γ =(V,E) consists of vertices v ∈ V connected by edges e ∈ E being intervals of the length l e on the real line R. The Laplace op- erator L(Γ )= − d 2 dx 2 acting in the Hilbert space of square integrable functions is unambiguously de- termined by the graph. The Laplace operator L(Γ ) is self-adjoint and has a discrete and non-negative spectrum [12]. If all graph vertices V have standard (called also natural, Kirchhoff, Neumann) boundary conditions, i.e. functions are continuous at vertices and the sums of their oriented derivatives at ver- tices are zero, then the Laplacian has a simple zero eigenvalue with the eigenfunction being a constant. When even one vertex V D of the graph has the Dirichlet boundary condition (functions is zero at the vertex) then spectral multiplicity of the eigen- value of 0 becomes zero instead of one, provided the graph is connected. The most important characteristic of metric graphs Γ =(V,E) are the total length L = ∑ e∈E l e and the Euler characteristic χ = |V |−|E|, where |V | and |E| denote the number of vertices and edges. The later one is a purely topological quantity, but as we have shown in [15–17] and [18], it can be also obtained from the graph spectrum. The microwave networks simulating quantum graphs [19–25], which is possible thanks to the for- mal analogy of the one-dimensional Schrödinger equation describing quantum graphs and the teleg- rapher’s equation for microwave networks [19, 22], were used in experiments to obtain graph spec- tra, while the pseudo orbits method developed in [26] was applied in numerical calculation of them. It should be emphasized that microwave networks are the only ones that allow experi- mental simulations of quantum systems with all three types of symmetry within the framework of the random matrix theory (RMT). These sym- metries are: Gaussian orthogonal ensemble (GOE) — systems with preserved time reversal symmetry (TRS) [17, 19–21, 23, 27, 28], Gaussian symplectic 525