Title: IsoNN: Isomorphic Neural Network for Graph Representation Learning and Classification. Abstract: Deep learning models have achieved huge success in numerous fields, such as computer vision and natural language processing. However, unlike such fields, it is hard to apply traditional deep learning models on the graph data due to the ‘node-orderless’ property. Normally, adjacency matrices will cast an artificial and random node-order on the graphs, which renders the performance of deep mod- els on graph classification tasks extremely erratic, and the representations learned by such models lack clear interpretability. To eliminate the unnecessary node- order constraint, we propose a novel model named Isomorphic Neural Network (ISONN), which learns the graph representation by extracting its isomorphic features via the graph matching between input graph and templates. ISONN has two main components: graph isomorphic feature extraction component and classification component. The graph isomorphic feature extraction component utilizes a set of subgraph templates as the kernel variables to learn the possible subgraph patterns existing in the input graph and then computes the isomorphic features. A set of permutation matrices is used in the component to break the node-order brought by the matrix representation. Three fully-connected layers are used as the classification component in ISONN. Extensive experiments are conducted on benchmark datasets, the experimental results can demonstrate the effectiveness of ISONN, especially compared with both classic and state-of-the-art graph classification methods.