Graph-based unsupervised feature selection algorithms have been shown to be promising for handling unlabeled and high-dimensional data. Whereas, the vast majority of those algorithms usually involve two independent processes, i.e., similarity matrix construction and feature selection. This incurs a poor similarity matrix that is obtained from original data, which retains constant for the following feature selection process and heavily affects the corresponding performance. Aiming to integrate these two processes into a unified framework, this paper proposes a novel unsupervised feature selection algorithm, named Graph Learning Unsupervised Feature Selection (GLUFS) to ensure the two processes proceed simultaneously. In particular, a new similarity matrix is derived from the original one, while the new matrix can adaptively maintain the manifold structure of data. Due to the fact that good individual features do not necessarily guarantee efficient combinations, the GLUFS algorithm adopts the ℓ2,0-norm sparsity constraint to achieve group feature selection. Eventually, we perform experiments on six public datasets with sufficient analysis, while the obtained results illustrate the effectiveness and superiority of our GLUFS over the considered algorithms.