The interconnection between brain regions in neurological disease encodes vital information for the advancement of biomarkers and diagnostics. Although graph convolutional networks are widely applied for discovering brain connection patterns that point to disease conditions, the potential of connection patterns that arise from multiple imaging modalities has yet to be fully realized. In this paper, we propose a multi-modal sparse interpretable GCN framework (SGCN) for the detection of Alzheimer's disease (AD) and its prodromal stage, known as mild cognitive impairment (MCI). In our experimentation, SGCN learned the sparse regional importance probability to find signature regions of interest (ROIs), and the connective importance probability to reveal disease-specific brain network connections. We evaluated SGCN on the Alzheimer's Disease Neuroimaging Initiative database with multi-modal brain images and demonstrated that the ROI features learned by SGCN were effective for enhancing AD status identification. The identified abnormalities were significantly correlated with AD-related clinical symptoms. We further interpreted the identified brain dysfunctions at the level of large-scale neural systems and sex-related connectivity abnormalities in AD/MCI. The salient ROIs and the prominent brain connectivity abnormalities interpreted by SGCN are considerably important for developing novel biomarkers. These findings contribute to a better understanding of the network-based disorder via multi-modal diagnosis and offer the potential for precision diagnostics. The source code is available at https://github.com/Houliang-Zhou/SGCN.