Network embedding has emerged as a promising research field for network analysis. Recently, an approach, named Barlow Twins, has been proposed for self-supervised learning in computer vision by applying the redundancy-reduction principle to the embedding vectors corresponding to two distorted versions of the image samples. Motivated by this, we propose Barlow Graph Auto-Encoder, a simple yet effective architecture for learning network embedding. It aims to maximize the similarity between the embedding vectors of immediate and larger neighborhoods of a node while minimizing the redundancy between the components of these projections. In addition, we also present the variational counterpart named Barlow Variational Graph Auto-Encoder. We demonstrate the effectiveness of our approach in learning multiple graph-related tasks, i.e., link prediction, clustering, and downstream node classification, by providing extensive comparisons with several well-known techniques on eight benchmark datasets.