During this work, we introduce the concept of hesitant fuzzy graph structure and explain it with an example. Some operations on hesitant fuzzy graph structures will be defined, that is, Cartesian product, composition, normal product, and tensor product. Further, we present the vertex degree in the products of hesitant fuzzy graph structure. Moreover, the connection is established between the vertex degree of hesitant fuzzy graph structures and the vertex degree of hesitant fuzzy graph structure attained by their distinct products.