Hi Malek,

thank you for your question. Well, fuzzy graphs and its variants have many real applications, for instance, Samanta and Pal, 2015, mentioned these examples: "data mining, image segmentation, clustering, image capturing, networking, communication, planning, scheduling, etc. Similarly, modeling of network topologies can be done using the concept of graphs. In addition, paths, walks, and circuits are used to solve many problems, viz., traveling salesman, database design, resource networking, etc."

Particularly, our research group is using fuzzy graphs for a semantic network implementation for text mining.

Reference: S. Samanta and M. Pal, "Fuzzy Planar Graphs," in IEEE Transactions on Fuzzy Systems, vol. 23, no. 6, pp. 1936-1942, Dec. 2015.

I hope that this answer might help your research.

Regards,

Fábio

Let (V, μ, ρ) be a fuzzy graph. We now provide two popular ways of defining the distance between a pair of vertices. One way is to define the “distance” dis(x,y) between x and y as the length of the shortest strongest path between them. This “distance” is symmetric and is such that dis(x,x) = 0 since by our definition of a fuzzy graph, no path from x to x can have strength greater than μ(x), which is the strength of the path of length 0. However, it does not satisfy the triangle property, as we see from the following example. Let V = {u, v, x, y,z}, ρ(x, u) = ρ(u, v) = ρ(v, z) = 1 and ρ(x, y) = ρ(y, z) = 0.5. Here any path from x to y or from y to z has strength ≤ 1/2 since it must involve either edge (x,y) or edge (y, z). Thus the shortest strongest paths between them have length 1. On the other hand, there is a path from x to z, through u and v, that has length 3 and strength 1. Thus dis(x,z) = 3 > 1 + 1 = dis(x,y) + dis(y, z) in this case.

https://link.springer.com/chapter/10.1007/978-3-7908-1854-3_3

Dear Fábio Lobato and Adriana Santos-Caballero

Thanks for your explanations.