7.3 Isomorphisms of Interval-Valued Fuzzy Graphs In this section, we consider various types of (weak) isomorphisms of interval-valued fuzzy graphs. Definition 7.3.1 Let G₁ = (A₁, B₁) and G₂ = (A2, B₂) be two interval-valued fuzzy graphs. A homomorphism f: G₁ G₂ is a mapping f: V₁ → V₂ such that for all x₁ € V₁, X₁y₁ € E₁, → (1) 15 (₁) 5 11 (f(xi)) ut ut (f(xi))
7.3 Isomorphisms of Interval-Valued Fuzzy Graphs In this section, we consider various types of (weak) isomorphisms of interval-valued fuzzy graphs. Definition 7.3.1 Let G₁ = (A₁, B₁) and G₂ = (A2, B₂) be two interval-valued fuzzy graphs. A homomorphism f: G₁ G₂ is a mapping f: V₁ → V₂ such that for all x₁ € V₁, X₁y₁ € E₁, → (1) 15 (₁) 5 11 (f(xi)) ut ut (f(xi))
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Transcribed Image Text:7.3 Isomorphisms of Interval-Valued Fuzzy Graphs
In this section, we consider various types of (weak) isomorphisms of interval-valued
fuzzy graphs.
Definition 7.3.1 Let G₁ = (A₁, B₁) and G₂ = (A₂, B₂) be two interval-valued
fuzzy graphs. A homomorphism f: G₁ → G₂ is a mapping f: V₁ → V₂ such
that for all x₁ € V₁, X₁y1 € E₁,
(i) μÃ, (x₁) ≤μÃ₂ (f(x₁)), µt, (x₂) ≤ μ₂ (f(x₁)),
(ii) μB₁ (x1Y1) ≤ µB₂ (ƒ (x₁) ƒ (y₁)), µg, (x₁y₁) ≤ µ₂ (ƒ (x₁) ƒ (yi)).
A bijective homomorphism with the property
(iii) μA, (x₁) = μ₂ (f(x₁)), µ₁ (x₁) = μ₂ (f(x₂))
is called a weak isomorphism and a weak co-isomorphism if
PB₂ (f (x₁) ƒ (y₁)), µF, (x₁₁)
=
(iv) μg, (x1y1)
=
X1, Y1 € V₁.
A bijective mapping f: G₁ G₂ satisfying (iii) and (iv) is called an isomor-
phism.
A₁ =
Example 7.3.2 Let G₁ = (V₁, E₁) and G₂ = (V2, E2) be graphs such that V₁ =
{a₁, b₁}, V₂ = {a2, b₂), E1 {a,b), E₂ = {a2b2}. Let A₁, A2, B₁, and B₂ be
interval-valued fuzzy subsets defined by
A₂ =
a1 b₁
0.2 0.3
bi
·(0.5.0.6)).
0.5 0.6
a2
((0.3-02).
a2 b2
0.6' 0.5
(f(x₁)f(yi)) for all
B₁ =
B₂ =
aby ayby
0.1 0.3
a₂b₂ a₂b₂
0.1 0.4
Then it follows that G₁ = (A₁, B₁) and G₂ = (A2, B₂) are interval-valued fuzzy
graphs of G and G₂, respectively. The map f: V₁
V₂ defined by f(a₁) =b₂
and f(b₁) = a₂ is a weak isomorphism, but it is not an isomorphism.
Now I want more examples of the subject
Isomorphisms of interval-Valued fuzzy graphs
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