Binary LCD Codes and Their Graph Representations

We establish a bijection between binary even LCD codes and simple graphs whose adjacency matrices are idempotent over $\FF_2$. This bijection preserves equivalence: permutation equivalence of codes corresponds exactly to graph isomorphism. Based on this framework, we characterize which distance-regular graphs yield LCD codes via intersection array parameters, prove that non-isomorphic conference graphs yield inequivalent codes, and classify LCD-derived graphs of small orders.