The Euler characteristic chi = vertical bar V vertical bar - vertical bar E vertical bar and the total length L are the most important topological and geometrical characteristics of a metric graph. Here vertical bar V vertical bar and vertical bar E vertical bar denote the number of vertices and edges of a graph. The Euler characteristic determines the number beta of independent cycles in a graph while the total length determines the asymptotic behavior of the energy eigenvalues via 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 lambda(1 ), ..., lambda(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 beta <= 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 chi can be used as a sensitive revealer of the fully connected graphs.