Large language models (LLMs) have shown strong performance in zero-shot summarization, but often struggle to model document structure and identify salient information in long texts. In this work, we introduce StrucSum, a training-free prompting framework that enhances LLM reasoning through sentence-level graph structures. StrucSum injects structural signals into prompts via three targeted strategies: Neighbor-Aware Prompting (NAP) for local context, Centrality-Aware Prompting (CAP) for importance estimation, and Centrality-Guided Masking (CGM) for efficient input reduction. Experiments on ArXiv, PubMed, and Multi-News demonstrate that StrucSum consistently improves both summary quality and factual consistency over unsupervised baselines and vanilla prompting. In particular, on ArXiv, it increases FactCC and SummaC by 19.2% and 8.0% points, demonstrating stronger alignment between summaries and source content. The ablation study shows that the combination of multiple strategies does not yield clear performance gains; therefore, structure-aware prompting with graph-based information represents a promising and underexplored direction for the advancement of zero-shot extractive summarization with LLMs.