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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))

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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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
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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