  We prove an algebraic version of the Gauge-Invariant Uniqueness Theorem, a
result which gives information about the injectivity of certain homomorphisms
between ${\mathbb Z}$-graded algebras. As our main application of this theorem,
we obtain isomorphisms between the Leavitt path algebras of specified graphs.
  From these isomorphisms we are able to achieve two ends. First, we show that
the $K_0$ groups of various sets of purely infinite simple Leavitt path
algebras, together with the position of the identity element in $K_0$,
classifies the algebras in these sets up to isomorphism. Second, we show that
the isomorphism between matrix rings over the classical Leavitt algebras,
established previously using number-theoretic methods, can be reobtained via
appropriate isomorphisms between Leavitt path algebras.
