Timely detection of hardware vulnerabilities during the early design stage is critical for reducing remediation costs. Existing early detection techniques often require specialized security expertise, limiting their usability. Recent efforts have explored the use of large language models (LLMs) for Verilog vulnerability detection. However, LLMs struggle to capture the structure in Verilog code, resulting in inconsistent detection results. To this end, we propose VerilogLAVD, the first LLM-aided graph traversal rule generation approach for Verilog vulnerability detection. Our approach introduces the Verilog Property Graph (VeriPG), a unified representation of Verilog code. It combines syntactic features extracted from the abstract syntax tree (AST) with semantic information derived from control flow and data dependency graphs. We leverage LLMs to generate VeriPG-based detection rules from Common Weakness Enumeration (CWE) descriptions. These rules guide the rule executor that traversal VeriPG for potential vulnerabilities. To evaluate VerilogLAVD, we build a dataset collected from open-source repositories and synthesized data. In our empirical evaluation on 77 Verilog designs encompassing 12 CWE types, VerilogLAVD achieves an F1-score of 0.54. Compared to the LLM-only and LLM with external knowledge baselines, VerilogLAVD improves F1-score by 0.31 and 0.27, respectively.