Euler Characteristic of Graphs and Networks

The Euler characteristic chi = vertical bar V vertical bar - vertical bar E vertical bar is an important topological characteristic of graphs and networks. Here, vertical bar V vertical bar and vertical bar E vertical bar 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 lambda(1), ..., lambda(N) of a simple quantum graph. We will test this finding numerically, using chaotic graphs with vertical bar V vertical bar = 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 chi can be used to reveal whether the graph is planar or not.