Geodesies in Fuzzy Graphs

An arc (u, v) in a fuzzy graph H is called strong if its weight is at least as great as the strength of connectedness of u and v when (u, v) is deleted. A strong path is a path whose arcs are all strong, and a geodesic is a shortest strong path between its end nodes. Let S be a set of nodes. The closure (S) of S is the set of nodes that lie on geodesies between nodes of S. We say that S covers H if (S) = H. A minimal cover of H is called a basis. A (fuzzy) end node is a node that has only one strong neighbor. We prove that any cover of H contains all the end nodes of H, and that if H is a fuzzy tree, its set of end nodes is its unique basis, but not conversely. If H is connected, there is a strong path, and hence a geodesic, between any two nodes u, v of H. The length of a geodesic between u and v is called the g-distance dg(u, v). Using this concept of distance, we prove that the center of a fuzzy tree consists of either a single node or two nodes joined by a strong arc. A node is called a median of (u, v, w) if it lies on geodesies between u and v, v and w, and w and u. We prove that in a fuzzy tree, every triple of nodes has a unique median, but not conversely.