This paper proposes ERIC, an efficient framework for graph similarity computation (GSC) based on graph edit distance (GED). The key idea is to replace the computationally expensive node-level matching module in existing GSC models with a simple yet powerful regularization technique called Alignment Regularization (AReg). AReg imposes a node-graph correspondence constraint on the GNN encoder during training, enabling the model to capture fine-grained graph alignment without requiring explicit node-level matching. In the inference stage, the GNN encoder's graph-level representations are directly used to compute similarity scores, eliminating the need for AReg and significantly improving inference efficiency. To enhance the discriminative power of the learned representations, the paper introduces a multi-scale GED discriminator that captures both node-level and global-level graph similarities. The framework is evaluated on real-world datasets, demonstrating its effectiveness, efficiency, and transferability. ERIC achieves state-of-the-art performance on multiple benchmarks, outperforming existing methods in terms of accuracy and computational efficiency. The model's design allows for efficient inference without requiring explicit node-level interactions, making it suitable for large-scale graph similarity tasks. The paper also provides a detailed analysis of the theoretical foundations of GED in embedding space, showing that the optimal graph alignment can be inferred by minimizing the difference between intra-graph and cross-graph node-graph similarities. This insight motivates the design of AReg and the multi-scale GED discriminator, which together enable the model to capture fine-grained graph interactions without explicit node-level matching. The proposed framework is shown to be effective in various graph similarity tasks, including drug discovery, social network analysis, and program analysis.This paper proposes ERIC, an efficient framework for graph similarity computation (GSC) based on graph edit distance (GED). The key idea is to replace the computationally expensive node-level matching module in existing GSC models with a simple yet powerful regularization technique called Alignment Regularization (AReg). AReg imposes a node-graph correspondence constraint on the GNN encoder during training, enabling the model to capture fine-grained graph alignment without requiring explicit node-level matching. In the inference stage, the GNN encoder's graph-level representations are directly used to compute similarity scores, eliminating the need for AReg and significantly improving inference efficiency. To enhance the discriminative power of the learned representations, the paper introduces a multi-scale GED discriminator that captures both node-level and global-level graph similarities. The framework is evaluated on real-world datasets, demonstrating its effectiveness, efficiency, and transferability. ERIC achieves state-of-the-art performance on multiple benchmarks, outperforming existing methods in terms of accuracy and computational efficiency. The model's design allows for efficient inference without requiring explicit node-level interactions, making it suitable for large-scale graph similarity tasks. The paper also provides a detailed analysis of the theoretical foundations of GED in embedding space, showing that the optimal graph alignment can be inferred by minimizing the difference between intra-graph and cross-graph node-graph similarities. This insight motivates the design of AReg and the multi-scale GED discriminator, which together enable the model to capture fine-grained graph interactions without explicit node-level matching. The proposed framework is shown to be effective in various graph similarity tasks, including drug discovery, social network analysis, and program analysis.