Alzheimer's disease (AD) is known to be gradual in its progression of irreversible neuronal damage and eventual death. Patterns of atrophy are strongly correlated with AD pathology, specifically, morphological changes in brain shape, which have been identified to occur up to ten years before clinical diagnoses. Structural neuroimaging modalities (e.g., MRI) make it possible to analyze brain shape using intermediate representations of 3D shape such as voxels and point clouds but typically suffer from high computational complexity and an absence of smoothness in 3D shape. We propose geometric deep learning models for analyzing AD pathology using graph neural networks composed of fast and efficient spiral mesh convolutional layers, which are trained on surface mesh representations of neuroanatomical structures. Our discriminative spiral network outperforms alternative methods and shape representations for AD classification. Our proposed generative mesh networks, conditioned on AD diagnosis, demonstrate volume and surface area reductions in subcortical regions affected by AD neurodegeneration as well.