Isomorphism and Morita equivalence of graph algebras

For any countable graph E, we investigate the relationship between the Leavitt path algebra LC(E) and the graph C^*-algebra C^*(E). For graphs E and F, we examine ring homomorphisms, ring $*$-homomorphisms, algebra homomorphisms, and algebra $*$-homomorphisms between LC(E) and LC(F). We prove that in certain situations isomorphisms between LC(E) and LC(F) yield $*$-isomorphisms between the corresponding C^*-algebras C^*(E) and C^*(F). Conversely, we show that $*$-isomorphisms between C^*(E) and C^*(F) produce isomorphisms between LC(E) and LC(F) in specific cases. The relationship between Leavitt path algebras and graph C^*-algebras is also explored in the context of Morita equivalence.