This presentation will focus on developing mathematical and computational models that use the brain’s structural connectivity to predict the development of brain diseases, including Alzheimer’s, Parkinson’s, Huntington’s, ALS and other neurodegenerative diseases. I will first describe our original proposal that Alzheimer and other dementias are underpinned by misfolded pathologies that spread on the brain structural connectome. This process can be mathematically captured by the so-called "Network Diffusion Model". Several examples from AD, ALS, Huntington's, Parkinson's and other dementias will be demonstrated. I will then present new extensions of this model in many meaningful ways, incorporating protein aggregation, clearance, active axonal transport, and mediation by external genes, cells and neuroinflammation. Deep neural network implementations of these complex and computationally prohibitive models will be motivated, and preliminary work on physics-informed neural networks will be presented. I will also briefly describe recent work in modeling brain electrophysiology using similar graph spectral models. All above models centrally involve the brain’s complex network Laplacian eigen-spectrum and “graph harmonics.” Through this work, we have found significant differences in the model’s parameters that relate healthy brains to Alzheimer’s disease, sleep, epilepsy and infant brain maturation. The related papers will be briefly highlighted.