Large language models (LLMs) exhibit pronounced social biases. Output-level or data-optimization--based debiasing methods cannot fully resolve these biases, and many prior works have shown that biases are embedded in internal representations. We propose \underline{U}nified \underline{G}raph \underline{I}somorphism for \underline{D}ebiasing large language models (\textit{\textbf{UGID}}), an internal-representation--level debiasing framework for large language models that models the Transformer as a structured computational graph, where attention mechanisms define the routing edges of the graph and hidden states define the graph nodes. Specifically, debiasing is formulated as enforcing invariance of the graph structure across counterfactual inputs, with differences allowed only on sensitive attributes. \textit{\textbf{UGID}} jointly constrains attention routing and hidden representations in bias-sensitive regions, effectively preventing bias migration across architectural components. To achieve effective behavioral alignment without degrading general capabilities, we introduce a log-space constraint on sensitive logits and a selective anchor-based objective to preserve definitional semantics. Extensive experiments on large language models demonstrate that \textit{\textbf{UGID}} effectively reduces bias under both in-distribution and out-of-distribution settings, significantly reduces internal structural discrepancies, and preserves model safety and utility.