In this paper, we present a novel Siamese graph convolution network (GCN) for face sketch recognition. To build a graph from an image, we utilize a deep learning method to detect the image edges, and then use a superpixel method to segment the edge image. Each segmented superpixel region is taken as a node, and each pair of adjacent regions forms an edge of the graph. Graphs from both a face sketch and a face photo are input into the Siamese GCN for recognition. A deep graph matching method is used to share messages between cross-modal graphs in this model. Experiments show that the GCN can obtain high performance on several face photo-sketch datasets, including seen and unseen face photo-sketch datasets. It is also shown that the model performance based on the graph structure representation of the data using the Siamese GCN is more stable than a Siamese CNN model.