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While virtually all existing work on Automated Essay Scoring (AES) models an essay as a word sequence, we put forward the novel view that an essay can be modeled as a graph and subsequently propose GAT-AES, a graph-attention network approach to AES. GAT-AES models the interactions among essay traits in a principled manner by (1) representing each essay trait as a trait node in the graph and connecting each pair of trait nodes with directed edges, and (2) allowing neighboring nodes to influence each other by using a convolutional operator to update node representations. Unlike competing approaches, which can only model one-hop dependencies, GAT-AES allows us to easily model multi-hop dependencies. Experimental results demonstrate that GAT-AES achieves the best multi-trait scoring results to date on the ASAP++ dataset. Further analysis shows that GAT-AES outperforms not only alternative graph neural networks but also approaches that use trait-attention mechanisms to model trait dependencies.