Intuitionistic Fuzzy Soft Graph

Vol. 11, No.2, 2016, 63-77

ISSN: 2320 –3242 (P), 2320 –3250 (online) Published on 1 December 2016

www.researchmathsci.org

DOI: http://dx.doi.org/10.22457/ijfma.v11n2a2

63

International Journal of

Intuitionistic Fuzzy Soft Graph

A.M.Shyla1 and T.K.Mathew Varkey2

Kerala University Library, Palayam Thiruvananthapuram, Kerala Department of mathematics, TKM College of Engineering, Kollam, Kerala

1

Corresponding Author. Email: [email protected]; 2

[email protected]

Received 12 November 2016; accepted 25 November 2016

Abstract. We introduce the notions of intuitionistic fuzzy soft graph, strong intuitionistic

fuzzy soft graph, complete intutionistic fuzzy soft graph in this paper. Also studied about union of two Intuitionistic fuzzy soft graph and proved that the collection of intuitionistic fuzzy soft graph is closed under finite union.

Keywords: Intuitionistic fuzzy soft graph, strong intuitionistic fuzzy soft graph, complete

intuitionistic fuzzy soft graph, union of intuitionistic fuzzy soft graph.

AMS Mathematics Subject Classification (2010): 05C72

1. Introduction

Molodtsov [13] introduced the concept of soft set that can be seen as a new mathematical theory for dealing with uncertainties. Molodtsov applied this theory to several directions [13, 14, 15] and then formulated the notions of Soft number, Soft derivative, Soft integral, etc. in [16]. The soft set theory has been applied to many different fields with greatness. Maji [11] worked on theoretical study of soft sets in detail. The algebraic structure of soft set theory dealing with uncertainties has also been studied in more detail. Aktas and Cagman [2] introduced definition of soft groups, and derived their basic properties. The most appreciate theory to deal with uncertainties is the theory of fuzzy sets, developed by Zadeh [22] in 1965. But it has an inherent difficulty to set the membership function in each particular cases. The generalization of Zadeh’s fuzzy set called intuitionistic fuzzy set was introduced by Atanassov [4] which is characterized by a membership function and a non-membership function. In Zadeh’s fuzzy set, the sum of membership degree and non- membership degree is equal to one. In Atanassov’s intuitionistic fuzzy set the sum of membership degree and non- membership degree does not exeed one.

64

In 1736, Euler first introduced the concept of graph theory. The theory of graph is extremely useful tool for solving combinatorial problems in different areas such as geometry, algebra, number theory, topology, operation research, optimization and computer science, etc.. The first definition of fuzzy graphs was proposed by Kaffmann [8] in 1973, from Zadeh’s fuzzy relations [22]. But Rosenfeld [19] introduced another elaborated definition including fuzzy vertex and fuzzy edges and several fuzzy analogs of graph theoretic concepts. The first definition of intuitionistic fuzzy graph was introduced by Atanassov [5] in1999. Karunambigai and Parvathy [7] introduced intuitionistic fuzzy graph as a special case of Atanassov’s intuitionistic fuzzy graph. Soft graph was introduced by Thumbakara and George [18]. In 2015, Mohinta and Samanta [20] introduced the concept of fuzzy soft graph.

In this paper, our aim is to introduce the notions of intuitionistic fuzzy soft graph, strong intuitionistic fuzzy soft graph, complete intutionistic fuzzy soft graph. Also studied about union of two intuitionistic fuzzy soft graph and proved that the collection of intuitionistic fuzzy soft graph is closed under finite union.

2. Preliminaries

Definition 2.1. A fuzzy set of a base set V ={v₁,v₂,…,v_n} _{(non-empty set) is specified by}

its membership function σ ; where

σ

:V →[0,1] assigning to each vi∈V , the degree

or grade to which V∈

σ

.

Definition 2.2. A fuzzy graph G=(

σ

,

µ

) is a pair of function

σ

:V →[0,1] and [0,1]

:V×V →

µ

, where for all v_i,v_j∈V we have

µ

(v_i,v_j)≤

σ

(v_i)∧

σ

(v_j) for

each (vi,vj)∈V×V. Here

σ

and

µ

are respectively called fuzzy vertex and fuzzy edge of the fuzzy graph G=(

σ

,

µ

).

Definition 2.3. Let G₁=(

σ

₁,

µ

₁) and G₂ =(

σ

₂,

µ

₂) be two fuzzy graphs over the set

V

. Then the

Union

of G₁ and G₂ is another fuzzy graph G₃=(

σ

₃,

µ

₃) over the set

V

,where

σ

3=

σ

1∨

σ

2 =max{

σ

1(vi),

σ

2(vi)} for every vi∈Vandi=1,2…n and

)} , ( ), , ( { = ) ,

( 1 2

3 vi vj max

µ

vi vj

µ

vi vj

µ

for every vi,vj∈Vandi, j=1,2…n

Definition 2.4. Let V be a non-empty set. An intuitionistic fuzzy set A in V defined as }

| )) ( ), ( , {(

= V v v v V

A

µ

A

γ

A ∈ which is characterised by a membership function

[0,1]

:V →

A

µ

and the non-membership function

γ

A:V →[0,1] and satisfying 1.

V v v

v A

A + ≤ ∈

≤ ( ) ( ) 1forevery

0

µ

γ

.

2. 0≤

µ

A(v),

γ

A(v),

π

A(v)≤1foreveryv∈V. 3.

π

A(v)=1−

µ

A(v)−

γ

A(v).

65

Obviously if

π

A(v)=0foreveryvinV, then the intuitionistic fuzzy set A is just Zadeh’s fuzzy set.

Definition 2.5. An intuitionistic fuzzy graph is defined as G=(V,E,

µ

,

γ

) where

1.V ={v1,v2,…,vn}(non-empty set) such that

µ

1:V →[0,1] and

γ

1:V →[0,1] denote the degree of membership and non-membership of the element v_i∈V respectively and 0≤

µ

1(vi)+

γ

1(vi)≤1foreveryvi∈V,i=1,2…n

2. E⊂V×V where

µ

2:V×V →[0,1]and

γ

2:V×V →[0,1] are such that

(i)

µ

2(vi,vj)≤min{

µ

1(vi),

µ

1(vj)} (ii)

γ

₂(v_i,v_j)≤max{

γ

₁(v_i),

γ

₁(v_j)} and

(iii)0≤

µ

2(vi,vj)+

γ

2(vi,vj)≤1, 0≤

µ

2(vi,vj),

γ

2(vi,vj),

π

(vi,vj)≤1

where

π

(v_i,v_j)=1−

µ

₂(v_i,v_j)−

γ

₂(v_i,v_j)forevery(v_i,v_j)∈E,i, j=1,2,…,n

Let

U

be an initial universal set, P be a set of parameters, P_{(U) be the power set of}

U

and A⊆P.

Definition 2.6. A pair (F,A) is called a softset over

U

if and only if F is a

mapping of A into the set of all subsets of the set

U

.

Definition 2.7. A pair (F,A) is called fuzzy soft set over

U

, where Fis a mapping

given by F :A→IU; IUdenotes the collection of all fuzzy subsets of U;A⊆P.

Definition 2.8. A pair (F,A)

⌣

is called an intuitionistic fuzzy soft set over

U

, where F

⌣

is a mapping given by F:A→IFU

⌣

; IFU denotes the collection of all intuitionistic

fuzzy subsets of U;A⊆P.

Definition 2.9. Let V ={v1,v2,…,vn}(non-empty set),P(parameterset) andA⊆P. Also let

1.

ρ

:A→IU(V) (IU(V) denotes collection of all fuzzy subsets in

V

)

a

a

a֏

ρ

( )=

ρ

_(say),

a

∈

A

_and :V →[0,1]

a

ρ

,

) ,

(A

ρ

Fuzzy soft vertex.

2.

µ

:A→IU(V×V),(IU(V×V) denotes collection of all fuzzy subsets in

V

×

V

)

a

a

a֏

µ

( )=

µ

_(say), a∈A _and :V×V →[0,1]

a

µ

(vi,vj)֏

µ

a(vi,vj) )

,

(A

µ

Fuzzy soft edge.

66 )

( ) ( ) ,

( i j a i j

a v v

ρ

v

ρ

v

µ

≤ ∧ for each (vi,vj)∈V×V,

n j

i and A a every

for ∈ , =1,2,…, _.

3. Intuitionistic fuzzy soft graph

Definition 3.1. Let G=(V,E) be a simple graph, V ={v₁,v₂,…,v_n} _{(non-empty set),} V

V

E⊆ × , P(parameterset)and A⊆P.Also let

1.

µ

₁ is a membership function defined on

V

by

1:A IF (V)

U

→

µ

(IFU(V)denotes collection of all intuitionistic fuzzy subsets in

V

)

a֏

µ

₁(a)=

µ

_{1a (say),}

a

∈

A

_and : [0,1]

1a V →

µ

,v_i ֏

µ

_1a(v_i) (A,

µ

₁) Intuitionistic fuzzy soft vertex of membership function and

γ

1 is a non-membership function defined on

V

by

1:A IF (V)

U

→

γ

(IFU(V)denotes collection of all intuitionistic fuzzy subsets in V )

a֏

γ

1(a)=

γ

1a (say),

a

∈

A

and

γ

1a:V →[0,1]_,vi ֏

γ

1a(vi)

) ,

(A

γ

₁ Intuitionistic fuzzy soft vertex of non-membership function such that n

V v v

vi a i i

a( ) ( ) 1forevery ,=1,2…

0≤

µ

1 +

γ

1 ≤ ∈ and a∈A.

2.

µ

₂ is a membership function defined on E by

2:A IF (V V)

U ×

→

µ

( IFU(V×V) denotes collection of all intuitionistic fuzzy

subsets in E)

a֏

µ

2(a)=

µ

2a (say),

a

∈

A

and

µ

2a:V×V →[0,1],(vi,vj)֏

µ

2a(vi,vj)

γ

2 is the non membership function defined on E by

2:A IF (V V)

U _×

→

γ

( IFU(V×V) denotes collection of all intuitionistic fuzzy

subsets in

V

×

V

)

a֏

γ

2(a)=

γ

2a (say), a∈A and

γ

2a:V×V →[0,1],(vi,vj)֏

γ

2a(vi,vj)

where (A,

µ

₂),(A,

γ

₂) are Intuitionistic fuzzy soft edge of membership function and non- membership function satisfying

(i)

µ

_2a(v_i,v_j)≤min{

µ

_1a(v_i),

µ

_1a(v_j)} (ii)

γ

2a(vi,vj)≤max{

γ

1a(vi),

γ

1a(vj)} and (iii) 0≤

µ

_2a(v_i,v_j)+

γ

_2a(v_i,v_j)≤1,

A a and n j

i E v v forevery v

v v

vi j a i j i j

a ≤ ∈ ∈

≤ ( , ), ( , ) 1, ( , ) , , =1,2, ,

0

µ

2

γ

2 …

Then =( , ,( , 1),( , 1),( , 2),( , 2))

*

µ

γ

µ

γ

A A

A A

E V

G is said to be the intuitionistic

fuzzy soft graph (IFSG) and this IFSG is denoted by G*_A,V,E.

Example 3.1. Consider a simple graph G=(V,E) where V ={v1,v2,v3} and

)} , ( ), , ( ), , {(

= v1 v2 v2 v3 v1 v3

67

intuitionistic fuzzy soft graph (IFSG), G*_A,V,E =(V,E,(A,

µ

₁) , (A,

γ

1) , (A,

µ

2) ,

)) ,

(A

γ

2 is described in Table 1 and Figure 1.

1(a) 1(b)

a

1

µ

v1 v2 v3

1

a 0.8 0.6 0

2

a 0.9 0.65 0.5

3

a 0.2 0.75 0.8

1(c) 1(d)

a

2

µ

(v1,v2) (v1,v3) (v2,v3)

1

a 0.1 0 0

2

a 0.15 0.25 0.1

3

a 0.01 0.1 0.2

Table 1:

1(a) IFSG corresponding 1(b) IFSG corresponding 1(c) IFSG corresponding to the parameter a1 to the parameter a2 to the parameter a3

Figure 1:

Definition 3.2. An intuitionistic fuzzy soft graph G*=(V,E,(A,

µ

₁),(A,

γ

₁), (A,

µ

2), ))

,

(A

γ

₂ is said to be strong intuitionistic fuzzy soft graph if

)} ( ), ( { = ) ,

( 1 1

2a vi vj min

µ

a vi

µ

a vj

µ

and

γ

2a(vi,vj)=max{

γ

1a(vi),

γ

1a(vj)} for every

A a and E v

v_i, _j)∈ ∈

( and is said to be complete intuitionistic fuzzy soft graph if

)} ( ), ( { = ) ,

( 1 1

2a vi vj min

µ

a vi

µ

a vj

µ

and

γ

2a(vi,vj)=max{

γ

1a(vi),

γ

1a(vj)} for every

A a and V v

v_i, _j∈ ∈ .

Example 3.2. Consider a simple graph G=(V,E) where V ={v1,v2,v3} and

a

1

γ

v1 v2 v3

1

a 0.1 0.2 0

2

a 0.05 0.25 0.4

3

a 0.6 0.1 0.15

a

2

γ

(v1,v2) (v1,v3) (v2,v3)

1

a 0.1 0 0

2

a 0.2 0.1 0.2

3

68 )}

, ( ), , ( ), , {(

= v₁ v₂ v₂ v₃ v₁ v₃

E . Let A={a₁,a₂,a₃} be the parameter set. Then the

strong intuitionistic fuzzy softgraph, G_A*_,V,E =(V,E,(A,

µ

₁),(A,

γ

₁),(A,

µ

₂),(A,

γ

₂)) is

described in Table 2 and Figure 2.

2(a) 2(b)

a

1

µ

v1 v2 v3

1

a 0.2 0.3 0.4

2

a 0.6 0.05 0.35

3

a 0.5 0.2 0.1

2(c) 2(d)

Table 2:

2(a) Strong IFSG corresponding 2(b) Strong IFSG corresponding 2(c) Strong IFSG corresponding to the parameter a1 to the parameter a2 to the parameter a3

Figure 2:

Example 3.3. Consider a simple graph G=(V,E) where V ={v1,v2,v3} and

)} , ( ), , ( ), , ( ), , ( ), , ( ), , {(

= v₁ v₂ v₂ v₃ v₁ v₃ v₁ v₁ v₂ v₂ v₃ v₃

E . Let A={a₁,a₂,a₃} be the

parameter set. Then the complete intuitionistic fuzzy soft graph, G*_A,V,E =(V,E,(A,

µ

₁),

) ,

(A

γ

1 , (A,

µ

2), (A,

γ

2)) is described in Table 3 and Figure 3.

a

1

γ

v₁ v₂ v₃

1

a 0.4 0.3 0.55

2

a 0.1 0.2 0.25

3

a 0.2 0.1 0.2

a

2

γ

(v1,v2) (v2,v3) (v1,v3)

1

a 0.4 0.55 0.55

2

a 0.2 0.25 0.25

3

a 0.2 0.2 0.2

a

2

µ

(v₁,v₂) (v₂,v₃) (v₁,v₃)

1

a 0.2 0.3 0.2

2

a 0.05 0.05 0.35

3

69

3(a) 3(b)

a

1

µ

v1 v2 v3

1

a 0.1 0.2 0

2

a 0.2 0.1 0

3

a 0 0.1 0

3(c)

3(d)

Table 3:

3(a) Complete IFSG corresponding 3(b) Complete IFSG corresponding 3(c) Complete IFSG corresponding to the parameter a1 to the parameter a2 to the parameter a3

Figure 3:

Definition 3.3. Let G=(V,E) be a simple graph, P(parameterset). Also let

P B A E E E V V

V₁, ₂⊆ , ₁, ₂ ⊂ , , ⊆ and * =( ₁, ₁,( , ₁),( , ₁),

1 , 1

, V E A

µ

A

γ

G_{AV E}

)) , ( ), ,

(A

µ

2 A

γ

2 and =( , ,( , ),( , ),( , ),( , ))

' 2 '

2 '

1 '

1 2 2 *

2 , 2

, V E B

µ

B

γ

B

µ

B

γ

GBV E be two

intuitionistic fuzzy soft graphs with V1∩V2 ≠

φ

. Then the union of intuitionistic fuzzy soft

graphs *

2 , 2 , *

1 , 1 , *

3 , 3

,V E = AV E BV E

C G G

G ∪ with the condition ( , ) ( ' ( ))

1 2a vi vj max

µ

a vk

µ

≥ and

a

1

γ

v1 v2 v3

1

a 0.4 0.3 0

2

a 0.3 0.2 0

3

a 0 0.3 0

a

2

µ

(v₁,v₂) (v₂,v₃) (v₁,v₃) (v₁,v₁) (v₂,v₂) (v₃,v₃)

1

a 0.1 0 0 0.1 0.2 0

2

a 0.1 0 0 0.2 0.1 0

3

a 0 0 0 0 0.1 0

a

2

γ

(v₁,v₂) (v₂,v₃) (v₁,v₃) (v₁,v₁) (v₂,v₂) (v₃,v₃)

1

a 0.4 0.3 0.4 0.4 0.3 0

2

a 0.3 0.2 0.3 0.3 0.2 0

3

70 and P a V v v v every for v max v

v_{i j a k i j k}

a( , )≥ ( ( )) , , ∈ , ∈

' 1 2

γ

γ

n k j i any

for , , =1,2,…, _{is defined to be} =( , ,( , ),( , '),

1 ' 1 3 3 * 3 , 3 , ′ ′

_γ

µ

C C E V

GCV E

)) , ( ), ,

( 2'

' 2

′ ′

_γ

µ

C

C where

C

=

A

∪

B

, V3 =V1∪V2 and

71 B A a V V V V v v v v B A a V V V V v v v v B A a V V V V v v v v v v A B a V V V V v v v v A B a V V V V v v A B a V V V V v v v v B A a V V V V v v v v B A a V V V V v v B A a V V V V v v v v v v j i j i a j i j i a j i j i a j i a j i j i a j i j i j i a j i j i a j i j i j i a j i a ∩ ∈ × × ∈ ∩ ∈ × × ∈ ∩ ∈ × ∩ × ∈ ∧ ∈ × ∩ × ∈ ∈ × × ∈ ∈ × × ∈ ∈ × ∩ × ∈ ∈ × × ∈ ∈ × × ∈ ′ and ) ( | ) ( ) , ( forevery ) , ( = and ) ( | ) ( ) , ( forevery ) , ( = and ) ( ) ( ) , ( forevery ) , ( ) , ( = | and ) ( ) ( ) , ( forevery ) , ( = | and ) ( | ) ( ) , ( forevery 0 = | and ) ( | ) ( ) , ( forevery ) , ( = | and ) ( ) ( ) , ( forevery ) , ( = | and ) ( | ) ( ) , ( forevery 0 = | and ) ( | ) ( ) , ( forevery ) , ( = ) , ( 1 1 2 2 ' 2 2 2 1 1 2 2 2 1 1 ' 2 2 2 2 1 1 ' 2 2 2 1 1 1 1 2 2 ' 2 2 2 1 1 2 1 1 2 2 2 2 1 1 2 ' 2

γ

γ

µ

γ

γ

γ

γ

γ

γ

Example 3.4. Consider a simple graph G=(V,E) where V ={v1,v2,v3,v4,v5} and

)} , ( ), , ( ), , ( ), , {(

= v₁ v₂ v₁ v₃ v₃ v₄ v₃ v₅

E . Let V₁={v₁,v₂,v₃}, E₁={(v₁,v₂),(v₁,v₃)}

and V2 ={v3,v4,v5},E2={(v3,v4),(v3,v5)}. Let P={a1,a2,a3,a4} be the parameter

set. Also let A,B⊆P such that A={a₁,a₂,a₃} and B={a₂,a₃,a₄}. Let )) , ( ), , ( ), , ( ), , ( , , (

= 1 1 1 1 2 2

*

1 , 1

, V E A

µ

A

γ

A

µ

A

γ

GAV E be the IFSG described in Table 4 and

Figure 4 and =( , ,( , ),( , ),( , ),( , 2'))

' 2 ' 1 ' 1 2 2 * 2 , 2

, V E A

µ

A

γ

A

µ

A

γ

GBV E be the IFSG described

in Table 5 and Figure 5. Then the union of * 1 , 1 ,V E A

G and *

2 , 2 ,V E B

G is

)) , ( ), , ( ), , ( ), , ( , , (

= 2'

' 2 ' 1 ' 1 3 3 * 3 , 3 , ′ ′ ′ ′

_γ

_µ

_γ

µ

C C C

C E V

GCV E given by Table 6 and Figure 6

where C= A∪B, V₃=V₁∪V₂ and E₃ =E₁∪E₂. 4(a)

a

1

µ

v1 v2 v3

1

a 0.3 0.2 0.3

2

a 0.3 0.4 0.5

3

a 0.2 0.4 0.3

4(b)

a

1

γ

v₁ v₂ v₃

1

a 0.3 0.3 0.4

2

a 0.4 0.3 0.3

3

a 0.4 0.3 0.3

4(c)

a

2

µ

(v1,v2) (v1,v3)

1

a 0.2 0.3

2

a 0.3 0.3

3

a 0.2 0.2

4(d)

a

2

γ

(v1,v2) (v1,v3)

1

a 0.3 0.3

2

a 0.3 0.2

3

a 0.4 0.3

72

4(a) IFSG Corresponding 4(b) IFSG Corresponding 4(c) IFSG Corresponding to the parameter a1 to the parameter a2 to the parameter a3

Figure 4:

5(a) 5(b)

' 1a

µ

v3 v4 v5

2

a 0.1 0.2 0.1

3

a 0.1 0.2 0.2

4

a 0.1 0.1 0.2

5(c) 5(d)

' 2a

µ

(v₃,v₄) (v₃,v₅)

2

a 0.1 0.05

3

a 0.1 0.1

4

a 0.07 0.1

Table 5:

5(a) IFSG corresponding 5(b) IFSG corresponding 5(c) IFSG corresponding to the parameter a2 to the parameter a3 to the parameter a4

Figure 5:

' 1a

γ

v3 v4 v5

2

a 0.1 0.2 0.1

3

a 0.2 0.1 0.1

4

a 0.1 0.1 0.1

' 2a

γ

(v3,v4) (v3,v5)

2

a 0.2 0.1

3

a 0.1 0.2

4

73

6(a) 6(b)

' 1

′ a

µ

v1 v2 v3 v4 v5

1

a 0.3 0.2 0.3 0 0

2

a 0.3 0.4 0.5 0.3 0.1

3

a 0.2 0.4 0.3 0.2 0.2

4

a 0 0 0.1 0.1 0.2

6(c) 6(d)

' 2

′ a

µ

(v1,v2) (v1,v3) (v3,v4) (v3,v5)

1

a 0.2 0.3 0 0

2

a 0.3 0.3 0.1 0.05

3

a 0.2 0.2 0.1 0.1

4

a 0 0 0.07 0.1

Table 6:

Figure 6:

Proposition 3.4. Let =( , ,( , ),( , ),( , ),( , 2')) '

2 '

1 '

1 3 3 *

3 , 3 ,

′ ′

′

′

_γ

_µ

_γ

µ

C C C

C E V

GCV E be the union

' 1

′ a

γ

v1 v2 v3 v4 v5

1

a 0.3 0.3 0.4 0 0

2

a 0.4 0.3 0.1 0.2 0.1

3

a 0.4 0.3 0.2 0.1 0.1

4

a 0 0 0.1 0.1 0.1

' 2a′

γ

(v1,v2) (v1,v3) (v3,v4) (v3,v5)

1

a 0.3 0.3 0 0

2

a 0.3 0.2 0.2 0.1

3

a 0.4 0.3 0.1 0.2

4

74

of two intuitionistic fuzzy soft graphs * =( ₁, ₁,( , ₁),( , ₁),

1 , 1

, V E A

µ

A

γ

G_{AV E}

)) , ( ), ,

(A

µ

₂ A

γ

₂ and =( , ,( , ),( , ),( , ),( , 2'))

' 2 ' 1 ' 1 2 2 * 2 , 2

, V E B

µ

B

γ

B

µ

B

γ

GBV E . Then

* 2 , 2 , * 1 , 1 , * 3 , 3

,V E = AV E BV E

C G G

G ∪ is an Intuitionistic fuzzy soft graph.

Proof: Consider the case a∈A| B.

If vi∈V1|V2; by definition3.3 ( )= 1 ( ) '

1a vi

µ

a vi

µ

′

and ( )= 1 ( )

'

1a vi

γ

a vi

γ

′ 1 ) ( ) ( = ) ( ) (

0 1 1

' 1 '

1 + + ≤

≤ ⇒ ′ ′ i a i a i a i

a v

γ

v

µ

v

γ

v

µ

, since *

1 , 1 ,V E A

G is an IFSG.

If v_i∈V₂| V₁; by definition3.3 1'a(vi)=0

′

µ

and 1'a(vi)=0

′

γ

Then clearly 0 ( ) 1' ( ) 1

'

1 + ≤

≤ ′ ′

i a i

a v

γ

v

µ

.

Also, if v_i∈V₁∩V₂; by definition3.3 ( )= 1 ( ) '

1a vi

µ

a vi

µ

′

and ( )= 1 ( )

'

1a vi

γ

a vi

γ

′ 1 ) ( ) ( = ) ( ) (

0 1 1

' 1 '

1 + + ≤

≤ ⇒ ′ ′ i a i a i a i

a v

γ

v

µ

v

γ

v

µ

, since *

1 , 1 ,V E A

G is an IFSG.

Similarly, when a∈B|A in all cases0 ( ) 1' ( ) 1

'

1 + ≤

≤ ′ ′

i a i

a v

γ

v

µ

Consider the case when a∈A∩B

If v_i∈V₁|V₂ or v_i∈V₂| V₁ clearly by definition3.3 0≤

µ

_1a'′(v_i)+

γ

₁'′_a(v_i)≤1 Now if

a

∈

A

∩

B

and v_i∈V₁∩V₂,

) ( ) ( = )

( _{1 1}'

'

1a vi

µ

a vi

µ

a vi

µ

′ ∨

and

γ

₁'′_a(v_i)=

γ

_1a(v_i)∧

γ

₁'_a(v_i)

Consider the cases ( )= ( ) ( )= 1 ( )

' 1 1

'

1a vi

µ

a vi and

γ

a vi

γ

a vi

µ

′ ′ , ) ( = ) ( ) ( = )

( ₁' ₁' ₁'

'

1a vi

µ

a vi and

γ

a vi

γ

a vi

µ

′ ′

Since *

1 , 1 ,V E A

G and *

2 , 2 ,V E B

G are IFSG, clearly in these cases 0 ( ) 1' ( ) 1

'

1 + ≤

≤ ′ ′

i a i

a v

γ

v

µ

.

When ( )= 1 ( )

'

1a vi

µ

a vi

µ

′

and ( )= 1' ( )

'

1a vi

γ

a vi

γ

′

By the condition in the definition 3.3 γ_2a(v_i,v_j)≥γ₁'_a(v_i) foreveryv_i,v_j∈V V v v every for v v

v_{i a i j i j}

a ≤ ∈

⇒ _{( ) ( , ) ,}

2 '

1

γ

γ

.

Also we have

γ

_2a(v_i,v_j)≤max{

γ

_1a(v_i),

γ

_1a(v_j)} by definition 3.1. V v every for v v

vi j a i i

a ≤ ∈

⇒ _{( , ) ( )}

1

2

γ

γ

( ) 1 ( )

'

1a vi

γ

a vi

γ

≤ ⇒ 1 ) ( ) ( ) ( ) ( = ) ( ) (

0 1 1

' 1 1 ' 1 '

1 + + ≤ + ≤

≤ ∴ ′ ′ i a i a i a i a i a i

a v

γ

v

µ

v

γ

v

µ

v

γ

v

µ

Similarly, when

µ

₁'′_a(v_i)=

µ

₁'_a(v_i) and

γ

₁'′_a(v_i)=

γ

_1a(v_i)

By the condition in the definition 3.3

µ

_2a(v_i,v_j)≥

µ

₁'_a(v_i) foreveryv_i,v_j∈V V v v every for v v

v_{i a i j i j}

a ≤ ∈

⇒ _{( ) ( , ) ,}

2 '

1

µ

µ

.

Also we have

µ

2a(vi,vj)≤min{

µ

1a(vi),

µ

1a(vj)} by definition 3.1. V v v every for v v

v_{i j a i i j}

a ≤ ∈

⇒ _{( , ) ( ) ,} 1 2

µ

µ

) ( ) ( ₁ '

1a vi

µ

a vi

µ

≤

75 1 ) ( ) ( ) ( ) ( = ) ( ) (

0 1 1 1

' 1 '

1 '

1 + + ≤ + ≤

≤ ∴ ′ ′ i a i a i a i a i a i

a v

γ

v

µ

v

γ

v

µ

v

γ

v

µ

Now to check the condition 2 of definition 3.1.

By using the definition 3.3,in all cases clearly got the inequalities 2(i)and2(ii) of

definition 3.1.That is µ₂'′_a(v_i,v_j)≤min{µ₁'′_a(v_i),µ₁'′_a(v_j)} and P a and V v v every for v v max v

v_{i j a i a j i j}

a ≤ ∈ ∈

′ ′ ′ _{( , ) { ( ),} ' _{( )} ,} 1 ' 1 '

2

γ

γ

γ

Now to check the condition 2(iii) of definition 3.1

If a∈A|Band (v_i,v_j)∈(V₁×V₁)|(V₂×V₂)or (v_i,v_j)∈(V₁×V₁)∩(V₂×V₂). Then by definition 3.3

E v v every for v v v v and v v v

v_{i j a i j a i j a i j i j}

a ∈ ′ ′ _{( , )= ( , ) ( , )= ( , ) ( , )} 2 ' 2 2 '

2

µ

γ

γ

µ

. E v v every for v v v v v v v

v_{i j a i j a i j a i j i j}

a + + ≤ ∈

≤

⇒₀ ′ _{( , )} ′ _{( , )= ( , ) ( , ) 1) ( , )}

2 2

' 2 '

2

γ

µ

γ

µ

.

Also if a∈A|Band (vi,vj)∈(V2×V2)|(V1×V1).

Then by definition 3.3

µ

₂'′_a(v_i,v_j)=0and

γ

₂'′_a(v_i,v_j)=0.

Obviously, 0≤

µ

₂'′_a(v_i,v_j)+

γ

₂'′_a(v_i,v_j)≤1, Since * 1 , 1 ,V E A

G is an IFSG. Similarly if

) ( ) ( ) , ( ) ( | ) ( ) , (

|Aand v v V2 V2 V1 V1 or v v V1 V1 V2 V2

B

a∈ i j ∈ × × i j ∈ × ∩ × .

By definition 3.3

E v v every for v v v v and v v v

v_{i j a i j a i j a i j i j}

a ∈ ′ ′ ) , ( ) , ( = ) , ( ) , ( = ) ,

( ₂' ₂' ₂'

'

2

µ

γ

γ

µ

. 1 ) , ( ) , ( = ) , ( ) , (

0≤ ₂' + ₂' ₂' + ₂' ≤

⇒ ′ ′ j i a j i a j i a j i

a v v

γ

v v

µ

v v

γ

v v

µ

.

Also if a∈B|Aand (vi,vj)∈(V1×V1)|(V2×V2). Then by definition 3.3 0 = ) , ( 0 = ) ,

( ₂'

'

2a vi vj and a vi vj

′

′

_γ

µ

. Obviously 0≤

µ

₂'′_a(v_i,v_j)+

γ

₂'′_a(v_i,v_j)≤1, Since *

2 , 2 ,V E B

G is an IFSG. Now consider the case when

a

∈

A

∩

B

.

If a∈A∩B and (vi,vj)∈(V1×V1)|(V2×V2), by definition3.3,

1 ) , ( ) , ( = ) , ( ) , (

0≤

µ

₂'′_a v_i v_j +

γ

₂'′_a v_i v_j

µ

_2a v_i v_j +

γ

_2a v_i v_j ≤ .

Similarly if

a

∈

A

∩

B

and (vi,vj)∈(V2×V2)|(V1×V1) by definition3.3, 1 ) , ( ) , ( = ) , ( ) , (

0≤

µ

₂'′_a v_i v_j +

γ

₂'′_a v_i v_j

µ

₂'_a v_i v_j +

γ

₂'_a v_i v_j ≤

Now consider the case when

a

∈

A

∩

B

and (vi,vj)∈(V1×V1)∩(V2×V2) By definition 3.3

µ

₂'′_a(v_i,v_j)=

µ

_2a(v_i,v_j)or

µ

₂'_a(v_i,v_j) and

) , ( ) , ( = ) ,

( _{2 2}'

'

2a vi vj

γ

a vi vj or

γ

a vi vj

γ

′

. The condition 2(iii) is clear when

) , ( = ) , ( ₂ '

2a vi vj

µ

a vi vj

µ

′

and

γ

₂'′_a(v_i,v_j)=

γ

_2a(v_i,v_j) and also when

) , ( = ) ,

( ₂'

'

2a vi vj

µ

a vi vj

µ

′ _{and ( , )=} ' _{( , )}

2 '

2a vi vj

γ

a vi vj

γ

′

Now consider the case

µ

₂'′_a(v_i,v_j)=

µ

_2a(v_i,v_j) and

γ

₂'′_a(v_i,v_j)=

γ

₂'_a(v_i,v_j)

76 n

k j

i, , =1,2,…, _and ( , ) { ' ( )}

1 1a vi vj max

γ

a vk

γ

≥ for any i,j,k=1,2…,n_{. Also}

)} ( ), ( { )

,

( 1 1

2a vi vj max

µ

a vi

µ

a vj

µ

≤ foreveryvi,vj∈Vby definition 3.1.

V v v everyy for v v v

v v

v v

v_{i j a i j a i j a i j i j}

a + + ∈

≤

∴₀ ′ _{( , )} ′ _{( , )= ( , )} ' _{( , ) ,} 2

2 '

2 '

2

γ

µ

γ

µ

V v v everyy for v v

max v

v_{i j a i a j i j}

a + ∈

≤µ₂ ( , ) {γ₁' ( ),γ₁' ( )} ,

V v v everyy for v v

v_{i j a i i j}

a + ∈

≤µ₂ ( , ) γ₁' ( ) ,

V v v everyy for

v v v

vi j a i j i j

a + ∈

≤

µ

2 ( , )

γ

2 ( , ) , 1

≤

Similarly when µ₂'′_a(v_i,v_j)=µ₂'_a(v_i,v_j) and γ₂'′_a(v_i,v_j)=γ_2a(v_i,v_j) We know that µ₂'_a(v_i,v_j)≤max{µ₁'_a(v_i),µ₁'_a(v_j)} foreveryvi,vj∈V

V v v everyy for v v v

v v

v v

vi j a i j a i j a i j i j

a + + ∈

≤

∴ ′ ′

, )

, ( ) , ( = ) , ( ) , (

0 2

' 2 '

2 '

2 γ µ γ

µ

V v v everyy for v v v

v

max _{a i a j} + _{a i j i j}∈

≤ {µ₁' ( ),µ₁' ( )} γ₂ ( , ) ,

V v v everyy for v v

v_{i a i j i j}

a + ∈

≤µ₁' ( ) γ₂ ( , ) ,

V v v everyy for

v v v

vi j a i j i j

a + ∈

≤

µ

2 ( , )

γ

2 ( , ) , 1

≤

Thus in all cases *

2 , 2 , *

1 , 1 , *

3 , 3

,V E = AV E BV E

C G G

G ∪ is an Intuitionistic fuzzy soft graph.

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