Single-cell RNA-sequencing (scRNA-Seq) is widely used to reveal the heterogeneity and dynamics of tissues, organisms, and complex diseases, but its analyses still suffer from multiple grand challenges, including the sequencing sparsity and complex differential patterns in gene expression. We introduce the scGNN (single-cell graph neural network) to provide a hypothesis-free deep learning framework for scRNA-Seq analyses. This framework formulates and aggregates cell-cell relationships with graph neural networks and models heterogeneous gene expression patterns using a left-truncated mixture Gaussian model. scGNN integrates three iterative multi-modal autoencoders and outperforms existing tools for gene imputation and cell clustering on four benchmark scRNA-Seq datasets. In an Alzheimer's disease study with 13,214 single nuclei from postmortem brain tissues, scGNN successfully illustrated disease-related neural development and the differential mechanism. scGNN provides an effective representation of gene expression and cell-cell relationships. It is also a powerful framework that can be applied to general scRNA-Seq analyses.