Diffusion models have recently emerged as a new powerful machinery for generative artificial intelligence on graphs, with applications ranging from drug design to knowledge discovery. However, despite their high potential, most, if not all, existing graph diffusion models are limited in their ability to holistically describe the intrinsic higher-order topological graph properties, which obstructs model generalizability and adoption for downstream tasks. We address this fundamental challenge and extract the latent salient topological graph descriptors at different resolutions by leveraging zigzag persistence. We develop a new computationally efficient topological summary,zigzag spaghetti (ZS), which delivers the most inherent topological properties simultaneously over a sequence of graphs at multiple resolutions. We derive theoretical stability guarantees of ZS and present the first attempt to integratedynamic topological information into graph diffusion models. Our extensive experiments on graph classification and prediction tasks suggest that ZS has a high promise not only to enhance performance of graph diffusion models, with gains up 10\%, but also to substantially booster model robustness.