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Spectral graph convolutional neural networks (GCNNs) have been producing encouraging results in graph classification tasks. However, most spectral GCNNs utilize fixed graphs when aggregating node features, while omitting edge feature learning and failing to get an optimal graph structure. Moreover, many existing graph datasets do not provide initialized edge features, further restraining the ability of learning edge features via spectral GCNNs. In this paper, we try to address this issue by designing an edge feature scheme and an add-on layer between every two stacked graph convolution layers in GCNN. Both are lightweight while effective in filling the gap between edge feature learning and performance enhancement of graph classification. The edge feature scheme makes edge features adapt to node representations at different graph convolution layers. The add-on layers help adjust the edge features to an optimal graph structure. To test the effectiveness of our method, we take Euclidean positions as initial node features and extract graphs with semantic information from point cloud objects. The node features of our extracted graphs are more scalable for edge feature learning than most existing graph datasets (in one-hot encoded label format). Three new graph datasets are constructed based on ModelNet40, ModelNet10 and ShapeNet Part datasets. Experimental results show that our method outperforms state-of-the-art graph classification methods on the new datasets by reaching 96.56% overall accuracy on Graph-ModelNet40, 98.79% on Graph-ModelNet10 and 97.91% on Graph-ShapeNet Part. The constructed graph datasets will be released to the community.