Graph convolutional network (GCN) based on the brain network has been widely used for EEG emotion recognition. However, most studies train their models directly without considering network dimensionality reduction beforehand. In fact, some nodes and edges are invalid information or even interference information for the current task. It is necessary to reduce the network dimension and extract the core network. To address the problem of extracting and utilizing the core network, a core network extraction model (CWGCN) based on channel weighting and graph convolutional network and a graph convolutional network model (CCSR-GCN) based on channel convolution and style-based recalibration for emotion recognition have been proposed. The CWGCN model automatically extracts the core network and the channel importance parameter in a data-driven manner. The CCSR-GCN model innovatively uses the output information of the CWGCN model to identify the emotion state. The experimental results on SEED show that: (1) the core network extraction can help improve the performance of the GCN model; (2) the models of CWGCN and CCSR-GCN achieve better results than the currently popular methods. The idea and its implementation in this paper provide a novel and successful perspective for the application of GCN in brain network analysis of other specific tasks. The code is available at https://github.com/ykhdu/CWGCN-CCSR-GCN.