Proper functioning of connected and automated vehicles (CAVs) is crucial for the safety and efficiency of future intelligent transport systems. Meanwhile, transitioning to fully autonomous driving requires a long period of mixed autonomy traffic, including both CAVs and human-driven vehicles. Thus, collaboration decision-making for CAVs is essential to generate appropriate driving behaviors to enhance the safety and efficiency of mixed autonomy traffic. In recent years, deep reinforcement learning (DRL) has been widely used in solving decision-making problems. However, the existing DRL-based methods have been mainly focused on solving the decision-making of a single CAV. Using the existing DRL-based methods in mixed autonomy traffic cannot accurately represent the mutual effects of vehicles and model dynamic traffic environments. To address these shortcomings, this article proposes a graph reinforcement learning (GRL) approach for multi-agent decision-making of CAVs in mixed autonomy traffic. First, a generic and modular GRL framework is designed. Then, a systematic review of DRL and GRL methods is presented, focusing on the problems addressed in recent research. Moreover, a comparative study on different GRL methods is further proposed based on the designed framework to verify the effectiveness of GRL methods. Results show that the GRL methods can well optimize the performance of multi-agent decision-making for CAVs in mixed autonomy traffic compared to the DRL methods. Finally, challenges and future research directions are summarized. This study can provide a valuable research reference for solving the multi-agent decision-making problems of CAVs in mixed autonomy traffic and can promote the implementation of GRL-based methods into intelligent transportation systems. The source code of our work can be found at https://github.com/Jacklinkk/Graph_CAVs.