Graph-based multi-view clustering aiming to obtain a partition of data across multiple views, has received considerable attention in recent years. Although great efforts have been made for graph-based multi-view clustering, it remains a challenge to fuse characteristics from various views to learn a common representation for clustering. In this paper, we propose a novel Consistent Multiple Graph Embedding Clustering framework(CMGEC). Specifically, a multiple graph auto-encoder(M-GAE) is designed to flexibly encode the complementary information of multi-view data using a multi-graph attention fusion encoder. To guide the learned common representation maintaining the similarity of the neighboring characteristics in each view, a Multi-view Mutual Information Maximization module(MMIM) is introduced. Furthermore, a graph fusion network(GFN) is devised to explore the relationship among graphs from different views and provide a common consensus graph needed in M-GAE. By jointly training these models, the common latent representation can be obtained which encodes more complementary information from multiple views and depicts data more comprehensively. Experiments on three types of multi-view datasets demonstrate CMGEC outperforms the state-of-the-art clustering methods.