Hyperspectral band selection aims to identify an optimal subset of bands from hyperspectral images (HSIs). Most existing methods explore the relationships between pair-wise pixels in a fixed graph. However, the quality of the initial fixed graph may be influenced by noises and user-defined parameters that may not be optimal for HSI analysis. In this paper, we propose a graph learning based autoencoder (GLAE) to achieve unsupervised hyperspectral band selection. Using the relationships of pair-wise pixels within HSIs, GLAE constructs the initial graph to characterize the geometric structures of HSIs and then adjusts the graph to adapt the band selection process. To solve the proposed model, we intoduce an alternative optimization algorithm. Experiments and comparisons on three HSI datasets demonstrate that the proposed GLAE achieves better results over the state-of-the-art methods.