Claim verification is a long-standing and challenging task that demands not only high accuracy but also explainability of the verification process. This task becomes an emerging research issue in the era of large language models (LLMs) since real-world claims are often complex, featuring intricate semantic structures or obfuscated entities. Traditional approaches typically address this by decomposing claims into sub-claims and querying a knowledge base to resolve hidden or ambiguous entities. However, the absence of effective disambiguation strategies for these entities can compromise the entire verification process. To address these challenges, we propose Verify-in-the-Graph (VeGraph), a novel framework leveraging the reasoning and comprehension abilities of LLM agents. VeGraph operates in three phases: (1) Graph Representation - an input claim is decomposed into structured triplets, forming a graph-based representation that integrates both structured and unstructured information; (2) Entity Disambiguation -VeGraph iteratively interacts with the knowledge base to resolve ambiguous entities within the graph for deeper sub-claim verification; and (3) Verification - remaining triplets are verified to complete the fact-checking process. Experiments using Meta-Llama-3-70B (instruct version) show that VeGraph achieves competitive performance compared to baselines on two benchmarks HoVer and FEVEROUS, effectively addressing claim verification challenges. Our source code and data are available for further exploitation.