Autism spectrum disorder (ASD) is a specific brain disease that causes communication impairments and restricted interests. Functional connectivity analysis methodology is widely used in neuroscience research and shows much potential in discriminating ASD patients from healthy controls. However, due to heterogeneity of ASD patients, the performance of conventional functional connectivity classification methods is relatively poor. Graph neural network is an effective graph representation method to model structured data like functional connectivity. In this paper, we proposed a functional graph discriminative network (FGDN) for ASD classification. On the basis of pre-built graph templates, the proposed FGDN is able to effectively distinguish ASD patient from health controls. Moreover, we studied the size of training set for effective training, inter-site predictions, and discriminative brain regions. Discriminative brain regions were determined by the proposed model to investigate its applicability and biomarkers for ASD identification. For functional connectivity classification and analysis, FGDN is not only an effective tool for ASD identification but also a potential technique in neuroscience research.