Chapter 4
Fuzzy Graph and Fuzzy relation
4.1.1 Graph and Fuzzy Graph
Graph
 G  (V, E)
 V : Set of vertices(node or element)
 E : Set of edges
An edge is pair (x, y) of vertices in V.
Fuzzy Graph



 
~
~
G  V, E
V : set of vertices
E : fuzzy set of edges between vertices
4.1.2 Fuzzy graph and fuzzy relation
Example 4.1 (an example of fuzzy graph)
a1
MG
b1
b2
a1
0.8
0.2
a2
0.3
0.0
a3
0.7
0.4
a2
a3
0.8
0.2
0.3
b1
0.7
b2
0.4
Fig 4.1 Fuzzy graph
4.1.2 Fuzzy graph and fuzzy relation
Example 4.2
  : nonnegative real numbers.
 x   and y  
 R  {(x, y) | x  y}, R    .
y
x
Fig 4.1 Fuzzy graph y  x ( y closes to x)
4.1.2 Fuzzy graph and fuzzy relation
Example 4.3
 The darkness of color stands for the strength of relation in (a)
 Relation (a, b) is stronger than that of relation (a, c).
 The corresponding fuzzy graph is shown in (b).
 the strength of relation is marked by the thickness of line.
a
b
c
a
a
b
b
c
(a) Fuzzy relation R
c
(b) Fuzzy graph
Fig 4.3 Fuzzy relation and fuzzy graph
4.1.2 Fuzzy graph and fuzzy relation
Example 4.4
b1
B2
b3
a1
0.5
1.0
0.0
a2
0.0
0.0
0.5
a3
1.0
1.0
0.0
a1
b1
a2
b2
a3
b3
mapping function (A)
{a1}  {(b1, 0.5), (b2, 1.0)}
{a2}  {(b3, 0.5)}
{a3}  {(b1, 1.0), (b2, 1.0)}
{a1, a2}  {(b1, 0.5), (b2, 1.0), (b3, 0.5)}
Fig 4.4 Fuzzy graph
4.1.2 Fuzzy graph and fuzzy relation
Example 4.5 A picture and fuzzy relation
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
y
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
3
2
1
-3
-2
-1
0
1
2
3
x
-1
-2
-3
Fig 4.5 Fuzzy graph
Fig 4.6 Fuzzy graph(by coordinates)
4.1.2 Fuzzy graph and fuzzy relation
Example 4.6 A graph and a fuzzy graph
y
1
y
1
1
1
x
1
x
-1
-1
(a) Graph R(x, y)  x2 + y2  1
-1
(b) Graph R (x, y)  x2 + y2  1
Fig 4.7 Fuzzy graph
4.1.3 -cut of Fuzzy Graph
Example 4.7
Appling -cut operation on fuzzy graph,
for example A  {a, b, c}, R  A  A is defined as follows.
a
MR
a
b
c
a
1.0
0.8
0.4
b
0.0
0.4
0.0
c
0.8
1.0
0.0
b
c
4.1.3 -cut of Fuzzy Graph
a
a
M R0.4
a
b
c
a
1.0
1.0
1.0
b
0.0
1.0
c
1.0
1.0
c
b
c
b
a
M R0.8
a
b
c
a
1.0
1.0
0.0
0.0
b
0.0
0.0
0.0
c
1.0
1.0
c
b
M R1.0
a
b
c
a
1.0
0.0
0.0
0.0
b
0.0
0.0
0.0
0.0
c
0.0
1.0
0.0
4.1.3 -cut of Fuzzy Graph
Example 4.8
R(x, y) = x/2 + y  1
y
y
1
1
0.5
0
0.5
1
2
Fig 4.9 Graphical form of R
x
0
1
2
x
Fig 4.10 Graphical representation of R0.5
4.1.3 -cut of Fuzzy Graph
Example 4.9
A(x) = x
R(x,y) = x+y  1, x  A, 0  y  1
A(x)
y
1
1
0
Fig 4.11 Set A(x)= x
1
x
0
1
x
Fig 4.12 Relation A(x,y)= x+y  1 , xA
4.1.3 -cut of Fuzzy Graph
Example 4.10
A={ x | x close to 2k, k = -1,0, 1,2,….}
A(x) = Max[0, cosx].
 A ( x)
A(x)
0.5
1



2


3
0

3

2
Set A(x)=cosx  0

3
2
5
3
2
7
3
5
2
x



2


3
0

3

2

3
2
5
3
Fig 4.14 -cut set A0.5
2
7
3
5
2
x
4.1.3 -cut of Fuzzy Graph
y



2



0
3
3


3
2
2
2
5
3
7
3
5
2
x
Fig 4.15 Relation R(x,y)=cosx.
y
0.5



2


3
0

3

2

3
2
5
3
Fig 4.16 -cut relation R0.5
2
7
3
5
2
x
4.2 Characteristics of Fuzzy Relation
4.2.1 Reflexive Relation

For all x  A, if R(x, x)  1
Example 4.8 A  {2, 3, 4, 5}
R : For x, y  A, “x is close to y”


R
2
3
4
5
2
1.0
0.9
0.8
0.7
3
0.9
1.0
0.9
0.8
4
0.8
0.9
1.0
0.9
5
0.7
0.8
0.9
1.0
If  x  A, R(x, y)≠1, then the relation is called “ irreflexive ” .
If  x A, R(x, y)≠1, then it is called “ antireflexive ”
4.2.2 Symmetric Relation
Symmetric
 (x, y)  A  A
 R(x, y)    R(y, x) = 
Antisymmetric
 (x, y)  A  A, x  y
 R(x, y)  R(y, x) or R(x, y)  R(y, x)  0
asymmetric or nonsymmetic
 (x, y)  A  A, x  y
 R(x, y)  R(y, x)
Perfect antisymmetric
  (x, y)  A  A, x  y
 R(x, y)  0  R(y, x)  0
4.2.3 Transitive Relation
Definition
 (x, y), (y, x), (x, z)  A  A
 R(x, z)  Max
[Min(R(x, y), R(y, z))]
If we use the symbol  for Max and  for Min, the last condition
becomes
 R(x, z)   [R(x, y)  R(y, z)]
If the fuzzy relation R is represented by fuzzy matrix MR, we know
that left side in the above formula corresponds to MR and right one
to MR2. That is, the right side is identical to the composition of
relation R itself. So the previous condition becomes,
 MR  MR2 or R  R2
4.2.3 Transitive Relation
Transitive relation example
0.6
0.2
a
b
1
0.4
1
0.3
c
For (a, a), we have R (a, a)  R2(a, a)
For (a, b), R (a, b)  R2(a, b)
We see MR  MR2 or R  R2
0.3
Fig 4.20 Fuzzy relation (transitive relation)
4.3 Classification of Fuzzy Relation
4.3.1 Fuzzy Equivalence Relation
Definition(Fuzzy equivalence relation)
(1) Reflexive relation
 x  A  R(x, x)  1
(2) Symmetric relation
 (x, y)  A  A, R(x, y)    R(y, x)  
(3) Transitive relation
 (x, y), (y, z), (x, z)  A  A
R(x, z) ≥ Max[Min[R(x, y), R(y, z)]]
y
4.3 Classification of Fuzzy Relation
Example 4.12 (Graph of fuzzy equivalence relation )
a
b
c
d
a
a
1.0
0.8
0.7
1.0
1
1
0.8
0.7
b
0.8
1.0
0.7
0.8
c
0.7
0.7
1.0
0.7
d
1.0
0.8
0.7
1.0
1
b
0.8
d
0.7
0.7
c
1
1