Graph Laplacian methods are a central tool for analyzing data lying on a manifold. In this talk, we introduce the G-invariant graph Laplacian (G-GL), designed for manifold data closed under the action of a known matrix (Lie) group G. We present the G-GLs key properties and show that exploiting the symmetries in the data has significant theoretical and computational advantages. We then show how to use the G-GLs eigendecomposition to construct G-equivariant and G-invariant embeddings of the data, related to certain random walks over the data. Finally, we discuss applications to the cryo-electron microscopy class averaging problem.