In this paper, we deal with the issue of classifying normally distributed data in a high-dimensional setting when variables are more numerous than observations. Under a spar-sity assumption on terms of the inverse covariance matrix (the precision matrix), we adapt the method of the linear discriminant analysis (LDA) by including a sparse estimate of the precision matrix over all populations. Furthermore, we develop a variable selection procedure based on the graph associated to the estimated precision matrix. For that, we define a discriminant capacity for each connected components of the graph, and keep variables of the most discriminant components. The adapted LDA and its selection procedure are both evaluated on synthetic data, and applied to real data from PET brain images for the classification of patients with Alzheimer's disease.