This paper introduces Edge-Conditioned Convolution (ECC), a novel convolution operation designed for graph-structured data. ECC generalizes the standard convolution operator from regular grids to arbitrary graphs while avoiding spectral domain methods, allowing for handling graphs of varying size and connectivity. The key innovation is conditioning filter weights on specific edge labels in the neighborhood of a vertex, which enhances the representation of local graph neighborhoods. The authors demonstrate the effectiveness of ECC in point cloud classification, achieving state-of-the-art performance on the Sydney dataset, and in graph classification, outperforming other deep learning approaches on the NCI1 dataset. The method is also applied to general graphs, showing competitive performance. The paper includes a detailed formulation of ECC, experimental results, and discussions on related work, edge labeling, and extensions. The source code is available at <https://github.com/mys007/ecc>.This paper introduces Edge-Conditioned Convolution (ECC), a novel convolution operation designed for graph-structured data. ECC generalizes the standard convolution operator from regular grids to arbitrary graphs while avoiding spectral domain methods, allowing for handling graphs of varying size and connectivity. The key innovation is conditioning filter weights on specific edge labels in the neighborhood of a vertex, which enhances the representation of local graph neighborhoods. The authors demonstrate the effectiveness of ECC in point cloud classification, achieving state-of-the-art performance on the Sydney dataset, and in graph classification, outperforming other deep learning approaches on the NCI1 dataset. The method is also applied to general graphs, showing competitive performance. The paper includes a detailed formulation of ECC, experimental results, and discussions on related work, edge labeling, and extensions. The source code is available at <https://github.com/mys007/ecc>.