The rapid dissemination of unverified information through social platforms like Twitter poses considerable dangers to societal stability. Identifying real versus fake claims is challenging, and previous work on rumor detection methods often fails to effectively capture propagation structure features. These methods also often overlook the presence of comments irrelevant to the discussion topic of the source post. To address this, we introduce a novel approach: the Structure-Aware Multilevel Graph Attention Network (SAMGAT) for rumor classification. SAMGAT employs a dynamic attention mechanism that blends GATv2 and dot-product attention to capture the contextual relationships between posts, allowing for varying attention scores based on the stance of the central node. The model incorporates a structure-aware attention mechanism that learns attention weights that can indicate the existence of edges, effectively reflecting the propagation structure of rumors. Moreover, SAMGAT incorporates a top-k attention filtering mechanism to select the most relevant neighboring nodes, enhancing its ability to focus on the key structural features of rumor propagation. Furthermore, SAMGAT includes a claim-guided attention pooling mechanism with a thresholding step to focus on the most informative posts when constructing the event representation. Experimental results on benchmark datasets demonstrate that SAMGAT outperforms state-of-the-art methods in identifying rumors and improves the effectiveness of early rumor detection.