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The current success of Graph Neural Networks (GNNs) usually relies on loading the entire attributed graph for processing, which may not be satisfied with limited memory resources, especially when the attributed graph is large. This paper pioneers to propose a Binary Graph Convolutional Network (Bi-GCN), which binarizes both the network parameters and input node attributes and exploits binary operations instead of floating-point matrix multiplications for network compression and acceleration. Meanwhile, we also propose a new gradient approximation based back-propagation method to properly train our Bi-GCN. According to the theoretical analysis, our Bi-GCN can reduce the memory consumption by an average of ~31x for both the network parameters and input data, and accelerate the inference speed by an average of ~51x, on three citation networks, i.e., Cora, PubMed, and CiteSeer. Besides, we introduce a general approach to generalize our binarization method to other variants of GNNs, and achieve similar efficiencies. Although the proposed Bi-GCN and Bi-GNNs are simple yet efficient, these compressed networks may also possess a potential capacity problem, i.e., they may not have enough storage capacity to learn adequate representations for specific tasks. To tackle this capacity problem, an Entropy Cover Hypothesis is proposed to predict the lower bound of the width of Bi-GNN hidden layers. Extensive experiments have demonstrated that our Bi-GCN and Bi-GNNs can give comparable performances to the corresponding full-precision baselines on seven node classification datasets and verified the effectiveness of our Entropy Cover Hypothesis for solving the capacity problem.