Abstract. Let E be a directed graph and K be any field.

I will begin by defining the Leavitt path algebra of E with

coefficients in K, denoted by L_K(E), and by giving examples of L_K(E) for various directed graphs E. The K-algebra L_K(E) is generated by the vertices, edges, and ghost edges of E satisfying certain relations. So what do the elements of L_K(E) look like?

I will show what multiplication means between the vertices, edges, and ghost of edges of E, hence showing how to multiply arbitrary generating elements of L_K(E).