Recent advances in single-cell RNA sequencing (scRNA-seq) technologies provide a great opportunity to study gene expression at cellular resolution, and the scRNA-seq data has been routinely conducted to unfold cell heterogeneity and diversity. A critical step for the scRNA-seq analyses is to cluster the same type of cells, and many methods have been developed for cell clustering. However, existing clustering methods are limited to extract the representations from expression data of individual cells, while ignoring the high-order structural relations between cells. Here, we proposed a new method (GraphSCC) to cluster cells based on scRNA-seq data by accounting structural relations between cells through a graph convolutional network. The representation learned from the graph convolutional network, together with another representation output from a denoising autoencoder network, are optimized by a dual self-supervised module for better cell clustering. Extensive experiments indicate that GraphSCC model outperforms state-of-the-art methods in various evaluation metrics on both simulated and real datasets.