Recognition of discriminative neural signatures and regions corresponding to emotions are important in understanding the neuron functional network underlying the human emotion process. Electroencephalogram (EEG) is a spatial discrete signal. In this paper, in order to extract the spatio-temporal characteristics and the inherent information implied by functional connections, a multichannel EEG emotion recognition method based on phase-locking value (PLV) graph convolutional neural networks (P-GCNN) is proposed. The basic idea of the proposed EEG emotion recognition method is using PLV-based brain network to model multi-channel EEG features as graph signals and then perform EEG emotion classification based on this model. Different from the traditional graph convolutional neural networks (GCNN) methods, the proposed P-GCNN method uses the PLV connectivity of EEG signals to determine the mode of emotional-related functional connectivity, which is used to represent the intrinsic relationship between EEG channels in different emotional states. On this basis, the neural network is trained to extract effective EEG emotional features. We conduct extensive experiments on the SJTU emotion EEG dataset (SEED) and DEAP dataset. The experimental results demonstrate that novel framework can improve the classification accuracy on both datasets, but not so effective on DEAP as on SEED, in which with 84.35% classification accuracy for SEED, and the average accuracies of 73.31%, 77.03% and 79.20% are, respectively, obtained for valence, arousal, and dominance classifications on the DEAP database.