In this dissertation we discussed the basic concepts of double layered fuzzy graph, triple layered fuzzy graph, fuzzy distance, and spanning tree. Fuzzy graph theory was introduced by Ariel Rosenfeld in 1975. Though it is very Young, it has been growing fast and has numerous applications in various fields. During the same time Yeh and Bang have also introduced various concepts of connectedness in fuzzy graphs. The degree of a vertex in fuzzy graphs which are obtained from two given fuzzy graphs using operations were discussed by Nagoor Gani and Radha. Radha and Kumaravelintroduced the concept of degree of an edge and total degree of an edge in fuzzy graphs. We study the degree of an edge in fuzzy graphs which are obtained from two given fuzzy graphs using the operations of union and join. In general, the degree of an edge in union and join of two fuzzy graphs G₁ and G₂ cannot be expressed in terms of these in G₁ and G₂.