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Fuzzy Graph

A fuzzy graph is a pair G=(sigma,mu), where sigma is a fuzzy set on a vertex set V and mu is a fuzzy relation on V satisfying

mu(x,y)<=min{sigma(x),sigma(y)}
(1)

for all x,y in V (Rosenfeld 1975). For an undirected fuzzy graph, mu is also a symmetric relation.

The support, or underlying crisp graph, has vertex set and edge set

V^*={x in V:sigma(x)>0}
(2)
E^*={(x,y) in V×V:mu(x,y)>0}.
(3)

When both sigma and mu take values only in {0,1}, the fuzzy graph reduces to an ordinary graph.

See also

Fuzzy Set, Graph, Relation, Weighted Graph

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References

Rosenfeld, A. "Fuzzy Graphs." In Fuzzy Sets and Their Applications to Cognitive and Decision Processes. (Ed. L. A. Zadeh, K.-S. Fu, K. Tanaka, and M. Shimura). New York: Academic Press, pp. 77-95, 1975. https://doi.org/10.1016/B978-0-12-775260-0.50008-6.

Cite this as:

Weisstein, Eric W. "Fuzzy Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FuzzyGraph.html

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