Recently graph-based methods have been adopted for Abstractive Text Summarization. However, existing graph-based methods only consider either word relations or structure information, which neglect the correlation between them. To simultaneously capture the word relations and structure information from sentences, we propose a novel Dual Graph network for Abstractive Sentence Summarization. Specifically, we first construct semantic scenario graph and semantic word relation graph based on FrameNet, and subsequently learn their representations and design graph fusion method to enhance their correlation and obtain better semantic representation for summary generation. Experimental results show our model outperforms existing state-of-the-art methods on two popular benchmark datasets, i.e., Gigaword and DUC 2004.