Traditional statistical methods have limitations in clustering high dimensional single-cell RNA sequencing (scRNA-seq) data which contain many biological and technical zeros. In this study, we propose a hybrid model which combines a Gaussian mixture model with a variational graph autoencoder algorithm as an extendable end-to-end framework to improve the cell clustering performance on scRNA-seq data. The entire algorithm is optimized simultaneously by a designed loss function. We apply it to three labeled and three simulated scRNA-seq datasets. We consider adjusted Rand index, normalized mutual information, and Silhouette coefficient as performance metrics to evaluate the clustering performance of proposed method in comparison to four selected baseline models. The results show that the proposed method outperforms the baseline methods and has great stability in cell clustering.