Cooperative positioning (CP) is considered as a promising positioning method for multiple autonomous underwater vehicles (multi-AUVs), because CP is characterized by low cost and high precision. In this research, a novel autonomous underwater vehicle (AUV) CP algorithm is proposed to enhance the global localization accuracy of the follower AUV. However, in traditional CP algorithm, the positioning error is large under the condition that the outlier data exists in the observation, which happens commonly. So in this research, a novel CP algorithm based on the factor graph and maximum correntropy(FGMC) for AUV is proposed to enhance the global localization accuracy of the AUV. Different from the traditional algorithms, this presented FGMC-based CP algorithm implements mathematically the Bayes filter by converting the global function estimation problem into the local one. And furthermore, the maximum correntropy is used as the cost function in the factor graph to estimation problem, this can reduce the influence of outliers on positioning accuracy. FGMC based cooperative positioning algorithm is established to mathematically implement the Bayes filter by converting the global function estimation problem into local function estimation problem. Furthermore, the maximum correntropy is used as the cost function in the factor graph to estimate the variables. To demonstrate and verify the proposed algorithm, simulation and real tests in different scenarios are performed in this research. Compared with the traditional CP algorithms, the positioning error of the proposed FGMC cooperative positioning algorithm is obviously smaller than that of the other algorithms.