# Applications of Fuzzy Graphs

• John N. Mordeson
• Premchand S. Nair
Chapter
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 46)

## Abstract

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.

## Keywords

Hamiltonian Path Fuzzy Subset Fuzzy Relation Connected Subgraph Fuzzy Graph

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## Copyright information

© Springer-Verlag Berlin Heidelberg 2000

## Authors and Affiliations

• John N. Mordeson
• 1
• Premchand S. Nair
• 2
1. 1.Center for Research in Fuzzy Mathematics and Computer ScienceOmahaUSA
2. 2.Department of Mathematics and Computer ScienceCreighton UniversityOmahaUSA