We apply the concept of m-polar fuzzy sets to graph structures. We introduce certain concepts in m-polar fuzzy graph structures, including strong m-polar fuzzy graph structure, m-polar fuzzy Di-cycle, m-polar fuzzy Di-tree, m-polar fuzzy Di-cut vertex, and m-polar fuzzy Di-bridge, and we illustrate these concepts by several examples. We present the notions of ϕ-complement of an m-polar fuzzy graph structure and self-complementary, strong self-complementary, totally strong self-complementary m-polar fuzzy graph structures, and we investigate some of their properties.