The development of efficient reduced order models (ROMs) from a deep learning perspective enables users to overcome the limitations of traditional approaches [1,2]. One drawback of the approaches based on convolutional autoencoders is the lack of geometrical information when dealing with complex domains defined on unstructured meshes. The present work proposes a framework for nonlinear model order reduction based on Graph Convolutional Autoencoders (GCA) to exploit emergent patterns in different physical problems, including those showing bifurcating behavior, high-dimensional parameter space, slow Kolmogorov-decay, and varying domains [3]. Our methodology extracts the latent space's evolution while introducing geometric priors, possibly alleviating the learning process through up- and down-sampling operations. Among the advantages, we highlight the high-generalizability in the low-data regime, and the great speedup. Moreover, we will present a novel graph feedforward network (GFN), extending the GCA approach to exploit multifidelity data, leveraging graph-adaptive weights, enabling large savings, and providing computable error bounds for the predictions [4]. [1] Lee, K. and Carlberg, K.T. Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders. Journal of Computational Physics, 404, p. 108973, 2020. [2] Fresca, S., Dedé, L. and Manzoni, A. A Comprehensive Deep Learning-Based Ap- proach to Reduced Order Modeling of Nonlinear Time-Dependent Parametrized PDEs. Journal of Scientific Computing, 87(2), p. 61, 2021. [3] Pichi, F., Moya, B. and Hesthaven, J. S. A graph convolutional autoencoder approach to model order reduction for parametrized PDEs. arXiv:2305.08573, 2023. [4] Morrison, O., Pichi, F., and Hesthaven, J. S. Graph Feedforward Network: a nonlin- ear reduction strategy for multifidelity applications. In preparation, 2023.