  Leavitt path algebras are shown to be algebras of right quotients of their
corresponding path algebras. Using this fact we obtain maximal algebras of
right quotients from those (Leavitt) path algebras whose associated graph
satisfies that every vertex connects to a line point (equivalently, the Leavitt
path algebra has essential socle). We also introduce and characterize the
algebraic counterpart of Toeplitz algebras.
