The development of social media platforms has revolutionized the speed and manner in which information is disseminated, leading to both beneficial and detrimental effects on society. While these platforms facilitate rapid communication, they also accelerate the spread of rumors and extremist speech, impacting public perception and behavior significantly. This issue is particularly pronounced during election periods, where the influence of social media on election outcomes has become a matter of global concern. With the unprecedented number of elections in 2024, against this backdrop, the election ecosystem has encountered unprecedented challenges. This study addresses the urgent need for effective rumor detection on social media by proposing a novel method that combines semantic analysis with graph neural networks. We have meticulously collected a dataset from PolitiFact and Twitter, focusing on politically relevant rumors. Our approach involves semantic analysis using a fine-tuned BERT model to vectorize text content and construct a directed graph where tweets and comments are nodes, and interactions are edges. The core of our method is a graph neural network, SAGEWithEdgeAttention, which extends the GraphSAGE model by incorporating first-order differences as edge attributes and applying an attention mechanism to enhance feature aggregation. This innovative approach allows for the fine-grained analysis of the complex social network structure, improving rumor detection accuracy. The study concludes that our method significantly outperforms traditional content analysis and time-based models, offering a theoretically sound and practically efficient solution.