Despite the unprecedented success of applying Large Language Models (LLMs) to graph discriminative tasks such as node classification and link prediction, its potential for graph structure generation remains largely unexplored. To fill this crucial gap, this paper presents a systematic investigation into the capability of LLMs for graph structure generation. Specifically, we design prompts triggering LLMs to generate codes that optimize network properties by injecting domain expertise from network science. Since graphs in different domains exhibit unique structural properties captured by various metrics (e.g., clustering coefficient capturing triangles in social networks while squares reflecting road segments in transportation networks), we first evaluate the capability of LLMs to generate graphs satisfying each structural property in different domains. After that, we select the optimal property configurations and benchmark the graph structure generation performance of LLMs against established graph generative models across multiple domains. Our findings shed light on generating graph structures from an LLM perspective. Our code is publically available https://github.com/yuwvandy/LLM-GraphGen.