The spread of fake news has long been a social issue and the necessity of identifying it has become evident since its dangers are well recognized. In addition to causing uneasiness among the public, it has even more devastating consequences. For instance, it might lead to death during pandemics due to unverified medical instructions. This study aims to build a model for identifying fake news using graphs and machine learning algorithms. Instead of scanning the news content or user information, the research explicitly focuses on the spreading network, which shows the interconnection among people, and graph features such as the Eigenvector centrality, Jaccard Coefficient, and the shortest path. Fourteen features are extracted from graphs and tested in thirteen machine learning models. After analyzing these features and comparing the test result of machine learning models, the results reflect that propensity and centrality contribute highly to the classification. The best performing models reach 0.9913 and 0.9987 separately from datasets Twitter15 and Twitter16 using a modified tree classifier and Support Vector Classifier. This model can effectively predict fake news, prevent potential negative social impact caused by fake news, and provide a new perspective on graph feature selection for machine learning models.