Dear researchers,
why do we need fuzzy graphs in computer vision or information systems?
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.
Certainly, graphs theory has many advantages to studying certain issues in applied science. In which we can represent some pure structures, this representation makes the study of these structures more easy.
Unfortunately, as pure mathematicians we interested mostly in the structures building, and give the full description of these structures, all the tolls and realizeing the use of these structures by explaining the theoretical parts. We mainly leave a lots of applications issues.
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