Graph clustering is a fundamental task in network analysis, aimed at uncovering meaningful groups of nodes based on structural and attribute-based similarities. Traditional Nonnegative Matrix Factorization (NMF) methods have shown promise in clustering tasks by providing low-dimensional representations of data. However, most existing NMF-based approaches are highly sensitive to noise and outliers, leading to suboptimal performance in real-world scenarios. Additionally, these methods often struggle to capture the underlying nonlinear structures of complex networks, which can significantly impact clustering accuracy. To address these limitations, this paper introduces Robust Self-Supervised Symmetric NMF (R3SNMF) to improve graph clustering. The proposed algorithm leverages a robust principal component model to handle noise and outliers effectively. By incorporating a self-supervised learning mechanism, R3SNMF iteratively refines the clustering process, enhancing the quality of the learned representations and increasing resilience to data imperfections. The symmetric factorization ensures the preservation of network structures, while the self-supervised approach allows the model to adaptively improve its clustering performance over successive iterations. In addition, R3SNMF integrates a graph-boosting method to improve how relationships within the network are represented. Extensive experimental evaluations on various real-world graph datasets demonstrate that R3SNMF outperforms state-of-the-art clustering methods in terms of both accuracy and robustness.