Band selection, as a special case of the feature selection problem, tries to remove redundant bands and select a few important bands to represent the whole image cube. This has attracted much attention, since the selected bands provide discriminative information for further applications and reduce the computational burden. Though hyperspectral band selection has gained rapid development in recent years, it is still a challenging task because of the following requirements: 1) an effective model can capture the underlying relations between different high-dimensional spectral bands; 2) a fast and robust measure function can adapt to general hyperspectral tasks; and 3) an efficient search strategy can find the desired selected bands in reasonable computational time. To satisfy these requirements, a multigraph determinantal point process (MDPP) model is proposed to capture the full structure between different bands and efficiently find the optimal band subset in extensive hyperspectral applications. There are three main contributions: 1) graphical model is naturally transferred to address band selection problem by the proposed MDPP; 2) multiple graphs are designed to capture the intrinsic relationships between hyperspectral bands; and 3) mixture DPP is proposed to model the multiple dependencies in the proposed multiple graphs, and offers an efficient search strategy to select the optimal bands. To verify the superiority of the proposed method, experiments have been conducted on three hyperspectral applications, such as hyperspectral classification, anomaly detection, and target detection. The reliability of the proposed method in generic hyperspectral tasks is experimentally proved on four real-world hyperspectral data sets.