Summary: In this paper, we study the self-adjoint vertex conditions of the product of two second-order or fourth-order local differential operators on metric graph. Based on the self-adjointness of the product operator on closed interval \([a, b]\), the necessary and sufficient conditions which make the product operators be self-adjoint operators are obtained by using the self-adjoint vertex conditions of the higher-order differential operator on metric graph. In addition, the article gives the relations between the self-adjointness of the product operators and the self-adjointness of the original operators.