We present geometric deep least-squares Petrov-Galerkin (GD-LSPG) [1], a projection-based model-order reduction approach that leverages a novel graph autoencoder architecture and a nonlinear manifold least-squares Petrov-Galerkin [2] projection scheme. Like convolutional neural network (CNN) autoencoders, graph autoencoders provide a robust nonlinear mapping for advection-dominated flows. However, the graph autoencoder offers greater flexibility over CNN autoencoders, as it is readily extendable to perform projection-based model-order reduction on computational models with an unstructured mesh. Furthermore, we find that the graph autoencoder's latent state variables are interpretable through the lens of the nonlinear manifold least-squares Petrov-Galerkin scheme. In particular, the Jacobian of the graph autoencoder's decoder provides time-dependent mode shapes that can be interpreted similarly to proper orthogonal decomposition modes. The construction of the graph autoencoder architecture comprises a two-step process. In the first step, a hierarchy of reduced graphs is generated using a spectral clustering algorithm to emulate the compressive abilities of CNNs. In the second step, a message-passing operation is trained at each level of the hierarchy to emulate the filtering capabilities of CNNs. We benchmark GD-LSPG on a one-dimensional Burgers' model with a structured mesh. Second, we illustrate the flexibility of the method by applying it to two test cases that use a Riemann solver for the Euler equations and a two-dimensional unstructured mesh. The proposed method is more flexible than traditional CNN-based autoencoders and provides considerably more accurate solutions than traditional affine projections for very low-dimensional latent spaces.