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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) ≤ PA₂ (f(x₁)), μ, (x₁) ≤ µ₂ (f(x₁)), (ii) PB, (X1Y1) ≤ PB₂ (ƒ (x₁) ƒ (y₁)), µg, (x₁ y₁) ≤ PB₂ (f(x₁) ƒ (y₁)). A bijective homomorphism with the property (iii) μ (x₁) = μ₂ (f(x₁)), µμ₁ (x₁) = µ₂ (f(x₂)) is called a weak isomorphism and a weak co-isomorphism if PB₂ (f(x₁) f (y₁)), P, (x₁3₁) = (iv) HB, (X1Y1) X₁, Y₁ € V₁. A bijective mapping f: G₁ G₂ satisfying (iii) and (iv) is called an isomor- phism. Example 7.3.3 Let G₁ and G₂ be as in Example 7.3.2 and let A₁, A₂, B₁, and B₂ b interval-valued fuzzy subsets defined by A₁ b₁ a1 - ((1₂²) 03). (₁ 0²5)). B₁ = = 0.2/ 0.4' A₂ = (f(x₁)f(y)) for all a2 ((04-03) (05.06)) B₂ = ab abi 0.1 0.3 | a2b2a2b2 0.1 0.3 Then 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(a) = b₂ and f(b₁) = a₂ is weak co-isomorphism, but it is not an isomorphism. Now I want more examples of the subject Isomorphisms of interval-Valued fuzzy graphs

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) ≤ PA₂ (f(x₁)), μ, (x₁) ≤ µ₂ (f(x₁)), (ii) PB, (X1Y1) ≤ PB₂ (ƒ (x₁) ƒ (y₁)), µg, (x₁ y₁) ≤ PB₂ (f(x₁) ƒ (y₁)). A bijective homomorphism with the property (iii) μ (x₁) = μ₂ (f(x₁)), µμ₁ (x₁) = µ₂ (f(x₂)) is called a weak isomorphism and a weak co-isomorphism if PB₂ (f(x₁) f (y₁)), P, (x₁3₁) = (iv) HB, (X1Y1) X₁, Y₁ € V₁. A bijective mapping f: G₁ G₂ satisfying (iii) and (iv) is called an isomor- phism. Example 7.3.3 Let G₁ and G₂ be as in Example 7.3.2 and let A₁, A₂, B₁, and B₂ b interval-valued fuzzy subsets defined by A₁ b₁ a1 - ((1₂²) 03). (₁ 0²5)). B₁ = = 0.2/ 0.4' A₂ = (f(x₁)f(y)) for all a2 ((04-03) (05.06)) B₂ = ab abi 0.1 0.3 | a2b2a2b2 0.1 0.3 Then 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(a) = b₂ and f(b₁) = a₂ is weak co-isomorphism, but it is not an isomorphism. Now I want more examples of the subject Isomorphisms of interval-Valued fuzzy graphs

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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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 (ƒ (x₁)), µ†, (x₁) ≤ µ†₂ (f(x₁)), (ii) μB, (X1Y1) ≤ PB₂ (ƒ (x₁) ƒ (y₁)), µB₁ (x₁y₁) ≤ µ₂ (ƒ (x₁) ƒ (yi)). A bijective homomorphism with the property (iii) μÃ₁ (x₁) = μ₂ (ƒ (x₂)), µ₁ (x₁) = µ₂ (f(x₂)) is called a weak isomorphism and a weak co-isomorphism if (iv) HB, (x₁y₁) = EB,(f (xı) f (yi)), F(XIY) = (f(x₁)f(yi)) for all X₁, y₁ € V₁. A bijective mapping f: G₁ → G₂ satisfying (iii) and (iv) is called an isomor- phism. Example 7.3.3 Let G₁ and G₂ be as in Example 7.3.2 and let A₁, A2, B₁, and B₂ be interval-valued fuzzy subsets defined by A₁ = b₁ a₁ b₁ 0.4' 0.5 A₂ = 0.2 0.3 b.)). -((0403) (0.5 0.6 a2 b2 a2 b2 9 0.4' , B₁ = 9 a₁b₁a₁b₁ 0.1 0.3 | a2b2a2b2 0.1 0.3 B₂ = = (ª Then G₁ = (A₁, B₁) and G₂ = (A2, B₂) are interval-valued fuzzy graphs of G₁ and G₂, respectively. The map f: V₁ → V₂ defined by f(a) = b₂ and f(b₁) = a₂ is a weak co-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 (ƒ (x₁)), µ†, (x₁) ≤ µ†₂ (f(x₁)), (ii) μB, (X1Y1) ≤ PB₂ (ƒ (x₁) ƒ (y₁)), µB₁ (x₁y₁) ≤ µ₂ (ƒ (x₁) ƒ (yi)). A bijective homomorphism with the property (iii) μÃ₁ (x₁) = μ₂ (ƒ (x₂)), µ₁ (x₁) = µ₂ (f(x₂)) is called a weak isomorphism and a weak co-isomorphism if (iv) HB, (x₁y₁) = EB,(f (xı) f (yi)), F(XIY) = (f(x₁)f(yi)) for all X₁, y₁ € V₁. A bijective mapping f: G₁ → G₂ satisfying (iii) and (iv) is called an isomor- phism. Example 7.3.3 Let G₁ and G₂ be as in Example 7.3.2 and let A₁, A2, B₁, and B₂ be interval-valued fuzzy subsets defined by A₁ = b₁ a₁ b₁ 0.4' 0.5 A₂ = 0.2 0.3 b.)). -((0403) (0.5 0.6 a2 b2 a2 b2 9 0.4' , B₁ = 9 a₁b₁a₁b₁ 0.1 0.3 | a2b2a2b2 0.1 0.3 B₂ = = (ª Then G₁ = (A₁, B₁) and G₂ = (A2, B₂) are interval-valued fuzzy graphs of G₁ and G₂, respectively. The map f: V₁ → V₂ defined by f(a) = b₂ and f(b₁) = a₂ is a weak co-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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