This paper proposes VerilogLAVD, a novel LLM-based approach that effectively leverages the structural characteristics of Verilog code for early detection of hardware vulnerabilities. VerilogLAVD introduces the Verilog Property Graph (VeriPG), which provides a unified representation of Verilog code, combining AST-based syntactic information with semantic information derived from control flow and data dependency graphs. LLM is used to generate VeriPG-based detection rules from CWE descriptions, and these rules are used to explore VeriPG to identify potential vulnerabilities. Experimental results on 77 Verilog designs (including 12 CWE types) show that VerilogLAVD achieves an F1-score of 0.54, outperforming LLM-only and external knowledge-based LLM techniques by 0.31 and 0.27, respectively.