The notion of a fuzzy line graph of a fuzzy graph is introduced. We gi
ve a necessary and sufficient condition for a fuzzy graph to be isomor
phic to its corresponding fuzzy line graph. We examine when an isomorp
hism between two fuzzy graphs follows from an isomorphism of their cor
responding fuzzy line graphs. We give a necessary and sufficient condi
tion for a fuzzy graph to be the fuzzy line graph of some fuzzy graph.