Large Language Models (LLMs) face significant limitations when applied to large-scale graphs, struggling with context constraints and inflexible reasoning. We present GraphChain, a framework that enables LLMs to analyze complex graphs through dynamic sequences of specialized tools, mimicking human exploratory intelligence. Our approach introduces two key innovations: (1) Progressive Graph Distillation, a reinforcement learning mechanism that generates optimized tool sequences balancing task relevance with information compression, and (2) Structure-aware Test-Time Adaptation, which efficiently tailors tool selection strategies to diverse graph topologies using spectral properties and lightweight adapters without costly retraining. Experiments show GraphChain significantly outperforms prior methods, enabling scalable and adaptive LLM-driven graph analysis.