We present a set of novel, energy-based models built on top of graph neural networks (GNN-EBMs) to estimate the unnormalized density of a distribution of graphs. GNN-EBMs can generate graphs implicitly via MCMC sampling. We compare the performance of GNN-EBMs trained using 3 different estimators: pseudolikelihood, conditional noise contrastive estimation, and persistent contrastive divergence (PCD). We find that all 3 estimators result in models that generalize well, while models trained with PCD generate samples that are competitive with state-of-the-art baselines. Finally, we discuss the potential of GNN-EBMs beyond generation for diverse tasks such as semi-supervised learning and outlier detection.