Euler Characteristic of Graphs and Networks

M. Lawniczaka, P. Kurasovb, S. Baucha, M. Białousa, L. Sirkoa aInstitute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02668 Warsaw, Poland bDepartment of Mathematics, Stockholm University, S-106 91 Stockholm, Sweden

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The Euler characteristic i=|V|-|E| is an important topological characteristic of graphs and networks. Here, |V| and |E| denote the number of vertices and edges of a graph or a network. It has been shown in [Phys. Rev. E 101, 052320 (2020)] that the Euler characteristic can be determined from a finite sequence of the lowest eigenenergies λ1,...,λN of a simple quantum graph. We will test this finding numerically, using chaotic graphs with |V|=8 vertices. We will consider complete (fully connected) and incomplete realizations of 8-vertex graphs. The properties of the Euler characteristic will also be tested experimentally using the sequence of the lowest resonances of the 5-vertex microwave network. We will show that the Euler characteristic i can be used to reveal whether the graph is planar or not.

DOI:10.12693/APhysPolA.139.323 topics: quantum graphs, microwave networks, Euler characteristic