We define the Floquet spectrum of a quantum graph as the collection of all spectra of operators of the form $D=(-i\frac{\partial}{\partial x}+\alpha(\frac{\partial}{\partial x}))^2$ where $\alpha$ is a closed $1$-form. We show that the Floquet spectrum completely determines planar 3-connected graphs (without any genericity assumptions on the graph). It determines whether or not a graph is planar. Given the combinatorial graph, the Floquet spectrum uniquely determines all edge lengths of a quantum graph.