Multi-hop question answering (QA) requires models to chain intermediate conclusions while remaining cost-aware. Existing training-free self-reflective protocols (generate -> critique -> revise) improve accuracy but operate on a flat token stream: revisions cannot identify which prior step is unreliable, and early stopping cannot exploit structural convergence. We introduce GraphEvoResearcher, a training-free protocol that lifts EvoResearcher's self-reflective meta-reward onto an explicit reasoning graph - nodes are intermediate conclusions/evidence and edges encode support, contradict, and depends_on. A graph-aware meta-reward combines correctness, efficiency, reflection, diversity, and four structural signals (connectivity, acyclicity, branch coverage, path convergence) to decide whether to revise, backtrack, or stop. A graph-convergence early-stop terminates when evidence branches converge on a single conclusion. On a real LLM (DeepSeek), over GSM8K, MATH, and BBH (n=50 each), GraphEvoResearcher reaches 98%, 92%, and 82% respectively, the highest of all baselines, with the largest gains on the hardest set (MATH: +24 pp over single-shot, +12 pp over flat reflection), while keeping construction cost bounded at ~2 steps / 4.5-5 nodes. Our results indicate the structural signal of a reasoning graph is most valuable exactly where single-step reasoning is unreliable.

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