In this article we determine the structure of a twisted first cohomology group of the first homology of a trivalent graph with a coefficient associated with the quantum Clebsch-Gordan condition. As an application we give a characterization of a combinatorial property, the external edge condition, which is defined in the study of the Heisenberg representation on the module constructed in the framework of topological quantum field theory, i. e. TQFT-module.
