In this paper, we introduce the concept of fuzzy Bi-Magic labeling in graphs. We defined Fuzzy Bi-Magic labeling for cycle and star graph. Further we investigated the Properties of such labeling on these graphs. A fuzzy graph G:(σ,μ) is known as fuzzy Bi-Magic graph if there exists two bijective functions σ:F→0,1 and μ: V× V → 0,1 such that μ(u,v) < σ(u) ^σ(v) with the property that the sum of the labels on the vertices and the labels of their incident edges is one of the constants either k_1 or k_2, independent of the choice of vertex. We investigated that fuzzy Cycle graphs and fuzzy Star graphs are fuzzy Bi-Magic graphs. Further, some properties related to fuzzy bridge and fuzzy cut node have been discussed.
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