Abstract: The investigation of conserved quantities of differential equations dates back to Sophus Lie's investigations in the 1870s. His and Emmy Noether's fundamental result motivates the investigation of continuous and discrete symmetries in the context of evolution equations on infinite-dimensional Hilbert spaces. We will present two classes of symmetries arising in the theory of differential operators on metric graphs. This will yield a Frucht- type theorem for such so-called quantum graphs.