Graph C*-algebras are constructed using projections corresponding to the vertices of the graph, and partial isometries corresponding to the edges of the graph. Here, we use the gauge-invariant uniqueness theorem to first establish that the C*-algebra of a graph composed of a directed cycle with finitely many edges emitting away from that cycle is Mn+k(C(T)), where n is the length of the cycle and k is the number of edges emitting away. We use this result to establish the main results of the thesis, which pertain to maximally edge-colored directed graphs. We show that the C*-algebra of any finite maximally edge-colored directed graph is *Mn(C){ Mn(C(T))}k, where n is the number of vertices of the graph and k depends on the structure of the graph. Finally, we show that this algebra is in fact isomorphic to Mn(*C{ C(T)}k).