Hardware system customized toward the demands of graph neural network learning would promote efficiency and strong temporal processing for graph-structured data. However, most amorphous/polycrystalline oxides-based memristors commonly have unstable conductance regulation due to random growth of conductive filaments. And graph neural networks based on robust and epitaxial film memristors can especially improve energy efficiency due to their high endurance and ultra-low power consumption. Here, robust and epitaxial Gd: HfO2-based film memristors are reported and construct a weighted echo state graph neural network (WESGNN). Benefiting from the optimized epitaxial films, the high switching speed (20 ns), low energy consumption (2.07 fJ), multi-value storage (4 bits), and high endurance (10) outperform most memristors. Notably, thanks to the appropriately dispersed conductance distribution (standard deviation = 7.68 nS), the WESGNN finely regulates the relative weights of input nodes and recursive matrix to realize state-of-the-art performance using the MUTAG and COLLAB datasets for graph classification tasks. Overall, robust and epitaxial film memristors offer nanoscale scalability, high reliability, and low energy consumption, making them energy-efficient hardware solutions for graph learning applications.