The traffic assignment problem (TAP) is essential to efficient road network operation and significantly influences urban mobility and development. Traditional optimization algorithms typically rely on strict assumptions and iterative optimization methods, making them computationally intensive and inflexible. Deep learning methods, conversely, offer a promising alternative by effectively capturing heterogeneous and nonlinear traffic flow characteristics from diverse datasets. This study introduced a graph convolutional network (GCN)-based framework for the user equilibrium traffic assignment problem (UE-TAP). Specifically, the proposed GCN model learned the implicit relationships between origin-destination (OD) demand matrices and the resulting equilibrium traffic flows, providing efficient and reliable traffic flow estimations without iterative computations. Furthermore, to accommodate variations in network topology, an innovative deep learning approach based on network partitioning and subgraph training was introduced, significantly enhancing the scalability and adaptability of the model. Numerical experiments conducted on the Sioux-Falls and Eastern Massachusetts networks demonstrated that the proposed model achieved robust and high-accuracy estimations across diverse scenarios. In fixed-topology scenarios with random variations in OD demands and link capacities, the proposed model achieved $R^2$ of approximately 0.90. Even in scenarios with random link failures coupled with varying OD demands and capacities, the model maintained $R^2$ of around 0.84. Overall, the proposed methodology represented a significant advancement in solving UE-TAP, particularly in dynamic environments with evolving road network structures.