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We propose a novel graph neural network (GNN) architecture for jointly optimizing user association, base station (BS) beamforming, and reconfigurable intelligent surface (RIS) phase shift in a multi-RIS aided multi-cell network. The proposed architecture represents BSs and users as nodes in a bipartite graph where the same type of nodes shares the same neural networks for generating messages and updating its representations, allowing for distributed implementation. In addition, we utilize a composite reflected channel estimation integrated between layers of the GNN structure to significantly reduce the signaling overhead and complexity required for channel estimation in a multi-RIS network. To avoid BS overload, load balancing is regularized in the training of the GNN and we further develop a collision avoidance algorithm to ensure strict load balancing at every BS. Numerical results show that the proposed GNN architecture is significantly more efficient than existing approaches. The results further demonstrate its strong scalability with network size and achieving a throughput performance approaching that of a centralized traditional optimization algorithm, without requiring individual RIS-reflected channels estimation and without the need for re-training or fine-tuning.