Outline Graph and fuzzy graph Characteristics of fuzzy relations Types of fuzzy relations Graph and fuzzy graph Graph Graph and fuzzy graph Fuzzy graph V is fuzzy node Graph and fuzzy graph ID: 385469
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Slide1
Fuzzy graph and relationSlide2
Outline
Graph and fuzzy graph
Characteristics of fuzzy relations
Types of fuzzy relationsSlide3
Graph and fuzzy graph
GraphSlide4
Graph and fuzzy graph
Fuzzy graph
~
V :
is fuzzy node
~Slide5
Graph and fuzzy graph
Fuzzy graph and fuzzy relationSlide6
Graph and fuzzy graph
α
-cut of fuzzy graphSlide7
Graph and fuzzy graph
α
-cut of fuzzy graphSlide8
Graph and fuzzy graph
Fuzzy networkSlide9
Characteristics of fuzzy relations
Reflexive relationSlide10
Characteristics of fuzzy relations
Symmetric relation
(
(
(
)
)
)
(
)Slide11
Characteristics of fuzzy relations
An example of perfect
antisymmetricSlide12
Characteristics of fuzzy relations
Transitive relationSlide13
Characteristics of fuzzy relations
An example of transitive relationSlide14
Characteristics of fuzzy relations
Transitive closure
Slide15
Characteristics of fuzzy relations
An example of transitive closureSlide16
Types of fuzzy relation
Fuzzy equivalence relation
Reflexive relation
Symmetric relation
Transitive relationSlide17
Types of fuzzy relation
a
b
c
d
a
1.0
0.8
0.4
0.1
b
0.8
1.0
0.0
0.0
c
0.4
0.0
1.0
0.5
d
0.1
0.0
0.5
1.0
(
i
) Reflexive relation √
(ii) Symmetric relation √
(iii) Transitive relation
a
b
c
d
a
1.0
0.8
0.4
0.4
b
0.8
1.0
0.4
0.1
c
0.4
0.4
1.0
0.5
d
0.4
0.1
0.5
1.0
XSlide18
Types of fuzzy relation
Fuzzy equivalence relation applications
Partition of setsSlide19
Types of fuzzy relation
Fuzzy equivalence relation applications
Partition by α-cutSlide20
Types of fuzzy relation
Fuzzy equivalence relation applications
Set similar to element x
Set similar to d
Set similar to eSlide21
Types of fuzzy relation
Fuzzy compatibility relation
Reflexive relation
Symmetric relationSlide22
Types of fuzzy relation
Fuzzy pre-order relation
Reflexive relation
Transitive relation
a
b
c
a
1.0
1.0
0.4
b
0.0
1.0
0.3
c
0.0
1.0
1.0
a
b
c
a
1.0
1.0
0.4
b
0
1.0
0.3
c
0
1.0
1.0Slide23
Types of fuzzy relation
Fuzzy semi-pre-order or
nonreflexive
fuzzy pre-order relation
Reflexive relation
Transitive relation
a
b
c
a
0.2
0.6
0.3
b
0.0
0.6
0.3
c
0.0
0.6
0.3Slide24
Types of fuzzy relation
Anti-reflexive fuzzy pre-order relation
Anti-reflexive relation
Transitive relation
a
b
c
a
0.0
0.4
0.3
b
0.0
0.0
0.0
c
0.0
0.0
0.0
(
x,x
) = 0
Slide25
Types of fuzzy relation
Fuzzy order relation
Reflexive relation
Antisymmetric
relation
Transitive relation
Perfect fuzzy order
Slide26
Types of fuzzy relation
Fuzzy order relation
crisp orderSlide27
Types of fuzzy relation
Fuzzy order relation
Dominating class
Dominated classSlide28
Types of fuzzy relationSlide29
Types of fuzzy relation
Dissimilitude relation
Antireflexive
relation
Symmetric relation
Min-Max transitive relationSlide30
Types of fuzzy relation
ex
Min number = 0.2Slide31
Types of fuzzy relation
ex
a
b
c
d
a
0.0
0.4
0.1
0.0
b
0.4
0.0
0.2
1.0
c
0.1
0.2
0.0
0.3
d
0.0
1.0
0.3
0.0
(
d,c
) ; 0.0 v 0.1 = 0.1
1.0 v 0.2 = 1.0
0.3 v 0.0 = 0.3
0.0 v 0.3 = 0.3
Min number = 0.1
(
b,c
); 0.4 v 0.1 = 0.4
0.0 v 0.2 = 0.2
0.2 v 0.0 = 0.2
1.0 v 0.3 = 1.0
Min number = 0.2Slide32
Types of fuzzy relation
Fuzzy
morphism
Homomorphism
Fuzzy
homomorphismSlide33
Types of fuzzy relation
Fuzzy
morphism
Strong homomorphism
Fuzzy strong
homomorphismSlide34
Types of fuzzy relation
Fuzzy homomorphismSlide35
Types of fuzzy relation
Fuzzy strong homomorphism