In this paper, we have introduced the concept of double vertex max-fuzzy graph and complete double vertex max-fuzzy graph of a fuzzy graph which are analogous to the concepts double vertex graph and complete double vertex graph in crisp graph theory. We have studied the connected, effective and complete properties of these operations. We have obtained the degrees of vertices in the double vertex max-fuzzy graph and the complete double vertex max-fuzzy graph of a fuzzy graph. Also we have provided some examples and illustrations.