Automatic Text Summarization(ATS) is distinctly beneficial due to a vast amount of textual data and time-consuming manual summarization. In order to enhance ATS for single document in huge datasets, a new extractive graph framework - text extractive SUMmarization framework based on EDge information with COreference resolution EDCOSUM is proposed in this paper that relies on coreference resolution, adding edge information in word-level graph and a sentence-ranking strategy. EDCOSUM combines the graph-based and statistical-based extractive summarization methods. It is a general method for any document (not limited to a specific domain). Moreover, two ranking strategies(sentence and LSA ranking strategy) are proposed for sentence selection. A set of extensive experiments on CNN/Daily Mail and NEWSROOM are conducted for investigating the proposed method. The widely used automatic evaluation tool: Recall-Oriented Understudy for Gisting Evaluation(ROUGE) is utilized to evaluate EDCOSUM. Compared to the state-of-the-art ATS methods, EDCOSUM achieves a competitive result by improvements of over the highest scores in the literature for metrics ROUGE-1, ROUGE-2 and ROUGE-L respectively.