The density peak clustering (DPC) algorithm is capable of identifying clusters of any shape, but there are two obvious shortcomings. First, it struggles to accurately identify the cluster centers of sparse clusters in datasets with uneven density distribution. Second, its remaining points assignment strategy can easily lead to a chain reaction, resulting in incorrect data point classification. Therefore, this paper introduces a density peak clustering algorithm called BRPMK-DPC based on bidirectional representative points (BRP) and mutual $K $-nearest neighbors (MKNN). The algorithm initially develop a method to calculate local density using forward $ K $-nearest neighbor representative points and backward inverse $K $-nearest neighbor representative points. This method efficiently identifies the correct cluster centers in datasets with uneven density distribution. Additionally, a method for assigning remaining points based on mutual $K $-nearest neighbors is proposed. This method offers adaptability in the assignment process, avoiding the drawback of using fixed $K $-values in derivative DPC algorithms. Finally, the BRPMK-DPC algorithm is tested on artificially synthesized datasets and real datasets. The experimental results demonstrate that the proposed algorithm not only efficiently identifies cluster centers with uneven density but also outperform the other seven compared algorithms on most datasets.