In this paper, the concept of a fuzzy graph structure is introduced based on the concept of graph structure given by E. Sampathkumar, in Bull. Kerala Math. Assoc., Vol 3, No.2 (2006 December), 65123. New concepts like ρi − edge, ρi − path, ρi − cycle, ρi − tree, ρi − forest, fuzzy ρi − cycle, fuzzy ρi − tree, fuzzy ρi − forest etc. are introduced. Some results are obtained. We continue the stud...