We develop the bialgebraic structure based on the set of functional graphs, which generalize the case of the forests of rooted trees. We use noncommutative polynomials as generating monomials of the functional graphs, and we introduce circular and arborescent brackets in accordance with the decomposition in connected components of the graph of a mapping of {1, 2, …, n} in itself as in the frame of the discrete dynamical systems. We give applications fordifferential algebras and algebras of differential operators.