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To better exploit search logs and model users' behavior patterns, numerous click models are proposed to extract users' implicit interaction feedback. Most traditional click models are based on the probabilistic graphical model (PGM) framework, which requires manually designed dependencies and may oversimplify user behaviors. Recently, methods based on neural networks are proposed to improve the prediction accuracy of user behaviors by enhancing the expressive ability and allowing flexible dependencies. However, they still suffer from the data sparsity and cold-start problems. In this paper, we propose a novel graph-enhanced click model (GraphCM) for web search. Firstly, we regard each query or document as a vertex, and propose novel homogeneous graph construction methods for queries and documents respectively, to fully exploit both intra-session and inter-session information for the sparsity and cold-start problems. Secondly, following the examination hypothesis, we separately model the attractiveness estimator and examination predictor to output the attractiveness scores and examination probabilities, where graph neural networks and neighbor interaction techniques are applied to extract the auxiliary information encoded in the pre-constructed homogeneous graphs. Finally, we apply combination functions to integrate examination probabilities and attractiveness scores into click predictions. Extensive experiments conducted on three real-world session datasets show that GraphCM not only outperforms the state-of-art models, but also achieves superior performance in addressing the data sparsity and cold-start problems.