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Next Point-of-Interest (POI) recommendation is an essential part of the flourishing location-based applications, where the demands of users are not only conditioned by their recent check-in behaviors but also by the critical influence stemming from geographical dependencies among POIs. Existing methods leverage Graph Neural Networks with the aid of pre-defined POI graphs to capture such indispensable correlations for modeling user preferences, assuming that the appropriate geographical dependencies among POIs could be pre-determined. However, the pre-defined graph structures are always far from the optimal graph topology due to noise and adaptability issues, which may decrease the expressivity of learned POI representations as well as the credibility of modeling user preferences. In this paper, we propose a novel Adaptive Graph Representation-enhanced Attention Network (AGRAN) for next POI recommendation, which explores the utilization of graph structure learning to replace the pre-defined static graphs for learning more expressive representations of POIs. In particular, we develop an adaptive POI graph matrix and learn it via similarity learning with POI embeddings, automatically capturing the underlying geographical dependencies for representation learning. Afterward, we incorporate the learned representations of POIs and personalized spatial-temporal information with an extension to the self-attention mechanism for capturing dynamic user preferences. Extensive experiments conducted on two real-world datasets validate the superior performance of our proposed method over state-of-the-art baselines.