The Leavitt path algebra of a graph

For any row-finite graph E and any field K we construct the Leavitt path algebraL(E) having coefficients in K. When K is the field of complex numbers, then L(E) is the algebraic analog of the Cuntz-Krieger algebra C∗(E) described in [I. Raeburn, Graph algebras, in: CBMS Reg. Conf. Ser. Math., vol. 103, Amer. Math. Soc., 2005]. The matrix rings Mn(K) and the Leavitt algebras L(1,n) appear as algebras of the form L(E) for various graphs E. In our main result, we give necessary and sufficient conditions on E which imply that L(E) is simple.