Gearboxes operate in challenging environments, which leads to a heightened incidence of failures, and ambient noise further compromises the accuracy of fault diagnosis. To address this issue, we introduce a fault diagnosis method that employs singular value decomposition (SVD) and graph Fourier transform (GFT). Singular values, commonly employed in feature extraction and fault diagnosis, effectively encapsulate various fault states of mechanical equipment. However, prior methods neglect the inter-relationships among singular values, resulting in the loss of subtle fault information concealed within. To precisely and effectively extract subtle fault information from gear vibration signals, this study incorporates graph signal processing (GSP) technology. Following SVD of the original vibration signal, the method constructs a graph signal using singular values as inputs, enabling the capture of topological relationships among these values and the extraction of concealed fault information. Subsequently, the graph signal undergoes a transformation via GFT, facilitating the extraction of fault features from the graph spectral domain. Ultimately, by assessing the Mahalanobis distance between training and testing samples, distinct defect states are discerned and diagnosed. Experimental results on bearing and gear faults demonstrate that the proposed method exhibits enhanced robustness to noise, enabling accurate and effective diagnosis of gearbox faults in environments with substantial noise.