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The prevalence and perniciousness of fake news has been a critical issue on the Internet, which stimulates the development of automatic fake news detection in turn. In this paper, we focus on the evidence-based fake news detection, where several evidences are utilized to probe the veracity of news (i.e., a claim). Most previous methods first employ sequential models to embed the semantic information and then capture the claim-evidence interaction based on different attention mechanisms. Despite their effectiveness, they still suffer from two main weaknesses. Firstly, due to the inherent drawbacks of sequential models, they fail to integrate the relevant information that is scattered far apart in evidences for veracity checking. Secondly, they neglect much redundant information contained in evidences that may be useless or even harmful. To solve these problems, we propose a unified Graph-based sEmantic sTructure mining framework, namely GET in short. Specifically, different from the existing work that treats claims and evidences as sequences, we model them as graph-structured data and capture the long-distance semantic dependency among dispersed relevant snippets via neighborhood propagation. After obtaining contextual semantic information, our model reduces information redundancy by performing graph structure learning. Finally, the fine-grained semantic representations are fed into the downstream claim-evidence interaction module for predictions. Comprehensive experiments have demonstrated the superiority of GET over the state-of-the-arts.