Derivations of Leavitt path algebra

In this paper, we describe the $K$-module $H{H}^1(L_K(\mathrm{\Gamma }))$ of outer derivations of the Leavitt path algebra $L_K(\mathrm{\Gamma })$ of a row-finite graph $\mathrm{\Gamma }$ with coefficients in an associative commutative ring $K$ with unit. We give an explicit formula for every outer derivation of $L_K(\mathrm{\Gamma })$. We also describe the Lie algebra structure of outer derivations of the Toeplitz algebra and we prove that every derivation of the Leavitt path algebra can be extended to a derivation of the corresponding $C^{\ast }$-algebra.