With the increasing research of deep learning in the EEG field, it becomes more and more important to fully extract the characteristics of EEG signals. Traditional EEG signal classification prediction neither considers the topological structure between the electrodes of the signal collection device nor the data structure of the Euclidean space to accurately reflect the interaction between signals. Graph neural networks can effectively extract features of non-Euclidean spatial data. Therefore, this paper proposes a feature selection method for epilepsy EEG classification based on graph convolutional neural networks (GCNs) and long short-term memory (LSTM) cells. While enriching the input of LSTM, it also makes full use of the information hidden in the EEG signals. In the automatic detection of epileptic seizures based on neural networks, due to the strong non-stationarity and large background noise of the EEG signal, the analysis and processing of the EEG signal has always been a challenging research. Therefore, experiments were conducted using the preprocessed Boston Children’s Hospital epilepsy EEG dataset, and input it into the GCN-LSTM model for deep feature extraction. The GCN network built by the graph convolution layer learns spatial features, then LSTM extracts sequence information, and the final prediction is performed by fully connected and softmax layers. The introduced method has been experimentally proven to be effective in improving the accuracy of epileptic EEG seizure detection. Experimental results show that the average accuracy of binary classification on the CHB-MIT dataset is 99.39%, and the average accuracy of ternary classification is 98.69%.