When an algebra is graded by a group, any additive character of the group induces a diagonalizable derivation of the ring. This construction is studied in detail for the case of a path algebra modulo relations and its fundamental group. We describe an injection of the character group into the first cohomology group following Assem-de la Pe\~{n}a. Rather general conditions are determined, in this context, which guarantee that a diagonalizable derivation is induced from the fundamental group.