Derivations and Hochschild cohomology of zigzag algebras

Let $\mathrm{\Gamma }$ be a connected graph without loops, cycles or multiple edges and $Z(\mathrm{\Gamma })$ the corresponding zigzag algebra. Then every Jordan derivation of $Z(\mathrm{\Gamma })$ is a derivation. Moreover, we will prove that the dimension of 1th Hochschild cohomology group of $Z(\mathrm{\Gamma })$ is one by computing the dimensions of linear spaces spanned by derivations and inner derivations. This implies that the dimension of the 1th Hochschild cohomology group of each algebra derived equivalent to a zigzag algebra is 1.