ReGraph: Iterative Graph Reading for Open-Ended Reasoning with Large Language Models

Open-ended graph reasoning should allow Large Language Models (LLMs) to decide what to read from a graph as their reasoning unfolds. Yet existing approaches typically expose graph information either as retrieved text or as a fixed representation formed before reasoning begins, limiting how graph evidence can adapt to the evolving language-model state. We introduce ReGraph (**Re**ad **Graph**), an iterative graph-reading framework that interleaves graph access with intermediate LLM computation. ReGraph maintains a persistent graph memory and repeatedly queries it with graph-query states refined by preceding Transformer layers, enabling each read to depend on both the instruction and evidence acquired so far. A query-conditioned topology diffusion mechanism further couples semantic relevance with graph structure, while keeping the graph outside the LLM context and the pretrained LLM frozen. Experiments on six benchmarks, including two introduced in this work, demonstrate the effectiveness of ReGraph across open-ended graph reasoning settings. The source code is available [here](https://anonymous.4open.science/r/ReGraph-submit).