<p>Automated essay scoring (AES) aims to predict the score of an essay based on the quality of the writing. A convincing essay structure promotes good articulation and logical-semantic discourse between sentences, which is essential for integrating and understanding the context and should always be considered from the perspective of teacher grading. However, existing AES methods for structure learning are either genre-specific or structure-shallow, which cannot generalize well to different essay genres and uncover the latent rich structure in essays. In fact, examining the potential interactions between sentences can advance the construction and understanding of essay structure. Based on this motivation, we propose the adaptive structure-aware graph neural network to explore the structure-aware representation learning of the essay, achieving a more comprehensive feature to better represent the essay. Extensive experimental results on both Automated Student Assessment Prize and Chinese Argumentative Essays datasets demonstrate the effectiveness of our method. We also design experiments in different settings to show that our method works well on different essay genres and topics with only score labels.</p>