Hyperspectral image (HSI) classification is fundamental to a wide range of remote sensing applications, such as precision agriculture, environmental monitoring, and urban planning, because HSIs provide rich spectral signatures that enable the discrimination of subtle material differences. Deep learning approaches, including Convolutional Neural Networks (CNNs), Graph Convolutional Networks (GCNs), and Transformers, have achieved strong performance in learning spatial–spectral representations. However, these models often face difficulties in jointly modeling long-range dependencies, fine-grained local structures, and non-Euclidean spatial relationships, particularly when labeled training data are scarce. This paper proposes a Spectral–Spatial Graph Transformer Network (SSGTN), a dual-branch architecture that integrates superpixel-based graph modeling with Transformer-based global reasoning. SSGTN consists of four key components, namely (1) an LDA-SLIC superpixel graph construction module that preserves discriminative spectral–spatial structures while reducing computational complexity, (2) a lightweight spectral denoising module based on $1\times 1$ convolutions and batch normalization to suppress redundant and noisy bands, (3) a Spectral–Spatial Shift Module (SSSM) that enables efficient multi-scale feature fusion through channel-wise and spatial-wise shift operations, and (4) a dual-branch GCN-Transformer block that jointly models local graph topology and global spectral–spatial dependencies. Extensive experiments on three public HSI datasets (Indian Pines, WHU-Hi-LongKou, and Houston2018) under limited supervision (1% training samples) demonstrate that SSGTN consistently outperforms state-of-the-art CNN-, Transformer-, Mamba-, and GCN-based methods in overall accuracy, Average Accuracy, and the $\kappa$ coefficient. The proposed framework provides an effective baseline for HSI classification under limited supervision and highlights the benefits of integrating graph-based structural priors with global contextual modeling.