<p>Various real-time problems typically possess information about fuzzy and inverse fuzzy values on its graph modeling. To address such situations, in this article, the bivalued fuzzy graph which involves the combination of less membership grade and high membership grade on the edges has been introduced. Subsequently, various operations such as bivalued cartesian product, bivalued tensor product and bivalued union of bivalued fuzzy graph have been defined. Further, various theorems associated with the aforesaid operations are proved to be bivalued fuzzy graph. Furthermore, the complement of bivalued fuzzy graph is also established and also proved to be a bivalued fuzzy graph.</p>