We propose a class of functional dependencies for graphs, referred to as GFDs. GFDs capture both attribute-value dependencies and topological structures of entities, and subsume conditional functional dependencies (CFDs) as a special case. We show that the satisfiability and implication problems for GFDs are coNP-complete and NP-complete, respectively, no worse than their CFD counterparts. We also show that the validation problem for GFDs is coNP-complete. Despite the intractability, we develop parallel scalable algorithms for catching violations of GFDs in large-scale graphs. Using real-life and synthetic data, we experimentally verify that GFDs provide an effective approach to detecting inconsistencies in knowledge and social graphs.