Large Language Models (LLMs) face significant limitations when applied to large-scale graphs, struggling with context constraints and inflexible reasoning. We introduce GraphChain, a novel framework enabling LLMs to analyze large graphs by orchestrating dynamic sequences of specialized tools, mimicking human exploratory processes. GraphChain incorporates two core technical contributions: (1) Progressive Graph Distillation, a reinforcement learning approach that learns to generate tool sequences balancing task relevance and intermediate state compression, thereby overcoming LLM context limitations. (2) Structure-aware Test-Time Adaptation (STTA), a mechanism using a lightweight, self-supervised adapter conditioned on graph spectral properties to efficiently adapt a frozen LLM policy to diverse graph structures via soft prompts without retraining. Experiments show GraphChain significantly outperforms prior methods, enabling scalable and adaptive LLM-driven graph analysis.