Fuzzy Hypergraphs

Strength of Certain Fuzzy Graphs

“…A fuzzy graph G = (V, µ, σ) [3] is a nonempty set V together with a pair of functions µ : V −→ [0, 1] and σ : V × V −→ [0, 1] such that for all u, v ∈ V , σ(u, v) = σ(uv) ≤ µ(u) ∧ µ(v). We call µ the fuzzy vertex set of G and σ the fuzzy edge set of G. Here after we denote the fuzzy graph G(µ, σ) simply by G and the underlying crisp graph of G by G * (V, E) with V as vertex set and E = {(u, v) ∈ V × V : σ(u, v) > 0} as the edge set or simply by G * .…”

“…In that case we also say that u and v are adjacent in G. A fuzzy graph G is complete [3] if σ(uv) = µ(u)∧µ(v) for all u, v ∈ V . A fuzzy graph G is a strong fuzzy graph [3] if σ(uv) = µ(u)∧ µ(v), ∀uv ∈ E. The strength of a strong fuzzy complete graph is one [6]. Let G 1 (V 1 , µ 1 , σ 1 ) and G 2 (V 2 , µ 2 , σ 2 ) be two fuzzy graphs with the underlying crisp graphs…”

“…A path P of length n − 1 in a fuzzy graph G [3] is a sequence of distinct vertices v 1 , v 2 , v 3 , . .…”

“…Azriel Rosenfeld [5] defined fuzzy graph based on the definitions of fuzzy sets and fuzzy relations and developed the theory of fuzzy graphs in 1975. John N. Mordeson and Premchand S. Nair [3] introduced different types of operations on fuzzy graphs. Sheeba M.B.…”

Some more results on fuzzy k-competition graphs

“…An edge is non-trivial if µ(x, y) = 0. The fuzzy graph ξ = (V , τ, ν) is called a fuzzy subgraph [24] …”