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Objective: To propose a coupled convolutional and graph convolutional network (CCGCN) model for diagnosis of Alzheimer's disease (AD) and its prodromal stage., Methods: The disease-related brain regions generated by group-wise comparison were used as the input. The convolutional neural networks (CNNs) were used to extract disease-related features from different locations on brain magnetic resonance (MR) images. The generated features via the graph convolutional network (GCN) were processed, and graph pooling was performed to analyze the inherent relationship between the brain topology and the diagnosis task adaptively. Through ADNI dataset, we acquired the accuracy, sensitivity and specificity of the diagnosis tasks for AD and its prodromal stages, followed by an ablation study on the model structure., Results: The CCGCN model outperformed the current state-of-the-art methods and showed a classification accuracy of 92.5% for AD with a sensitivity of 88.1% and a specificity of 96.0%., Conclusions: Based on the structural and topological features of the brain MR images, the proposed CCGCN model shows excellent performance in AD diagnosis and is expected to provide important assistance to physicians in disease diagnosis.