Deep learning-based hyperspectral image (HSI) classification has significant applications in remote sensing scene understanding. Whole-image propagation classification methods can achieve high-precision classification at the full-pixel level with a significant efficiency advantage. However, such methods generally face challenges such as the smoothness of high-frequency details and difficulty in efficiently modeling deep features. To address these issues, this article proposes a fast multikernel graph convolutional network (FMGCN) for fast whole-image classification. The model designs a multikernel graph convolution (MKGC) algorithm to enhance the expression ability of deep spectral-spatial features. This algorithm generalizes the graph convolution process from a single kernel propagation process to a propagation process with an arbitrary number of kernels to more fully exploit spectral-spatial features in non-Euclidean spaces. In addition, a symmetric compressed refinement network (SCRN) is constructed to restore the weakened local structural information in the whole-image propagation, and an embedded edge compensation strategy (EECS) is combined to further strengthen the high-frequency features in the boundary regions. Based on this design, FMGCN is able to capture more comprehensive global dependency information in a single whole-image propagation, achieving a uniform balance of high accuracy and high efficiency. Experiments on four representative hyperspectral datasets show that FMGCN achieves the best classification performance, effectively alleviates over-smoothing in complex structural scenarios, and outperforms batch propagation models in terms of inference efficiency.