Operations of Bipolar Fuzzy Soft Graph

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Operations of Bipolar Fuzzy Soft Graph

M. Rajeshwari1*

Department of Mathematics Presidency University Bangalore, India.

  1. A. Venkatesp

    1. Murugesan2

      Department of Mathematics, Reva University, Bangalore, India.

      Department of Mathematics and Comp. Science, Myanmar Institute of Information Technology, Myanmar.

      AbstractNumerical demonstrating, investigation and processing of issues with uncertainty is one of the most sizzling regions in interdisciplinary research including connected arithmetic, computational insight and choice sciences. It is significant that uncertainty emerges from different areas has altogether different nature and can't be caught inside a solitary numerical system. In this paper we initiate bipolar fuzzy soft graphs, vertex and edge- induced soft graphs and some operation of bipolar soft fuzzy graph are explore.

      Keywords Bipolar fuzzy soft graphs, strong bipolar fuzzy soft graphs, complete bipolar fuzzy soft graphs, regular bipolar fuzzy soft graphs.

      1. INTRODUCTION

        Soft set theory was introduced by Molodtsov[1]. Later Feng, Liu and Fotea combined soft set with fuzzy set and rough set. In 1975[1] Rosenfeld initiated fuzzy graph theory. During the same time various concepts in connectedness with fuzzy graphs was introduced by Yeh and Bang [2]. The concept of bipolar fuzzy graphs was introduced by Akramin. In this paper we initiate bipolar fuzzy soft graphs, vertex and edge- induced soft graphs and some operation of bipolar soft fuzzy graph.

      2. PRELIMINARIES

        Definition 2.1. If P, then the fuzzy soft set over the

        fuzzy subsets of S. It is defined by

        F, A Fei

        i i i i i i

        i i i i i i

        Fe c , c , c : c U,e A

        Definition 2.4. An intersection of two bipolar fuzzy soft sets (F, A) and (G, B) is a bipolar fuzzy soft set (H, C), where C = A B /= and H : C BF U is defined by H (e) = F (e) G (e) e C and denoted by (H, C) = (F, A)¯ (G, B).

        Denition 2.5[4]: Union of two bipolar fuzzy soft sets over a common universe U is a bipolar fuzzy soft set (H, C), where C=AB and H : C BFU is dened by

        H(e) = F(e) if e A\B

        = G(e) if e B\A

        = F(e) G(e) if e A B

        Denoted as F, A~ G, B H , C

      3. MAIN RESULT

G , , , A is such that

G , , , A is such that

Definition 3.1. An bipolar fuzzy soft graph

~ ~ ~

G J P , N L P , N

  1. G = (, ) is a simple graph

  2. is a nonempty set of parameters

universe is a pair of M , A where M : A CU , CU is a gathering of fuzzy subset of U.

Definition 2.2. [1] Let S be the universal set and P is the set

~

c)

c)

d)

d)

J P , N

~

L P , N

, A is a bipolar fuzzy soft set over

, Ais a bipolar fuzzy soft set over

of parameter then M , P is called soft set over S where M: PP(S).

~

e)

e)

J P , N

~

,

,

L P , N

is a bipolar fuzzy (sub)graph of

G for all . That is

Definition 2.2 [15]: If P, then the fuzzy soft set over the ~ ~ ~

universe is a pair of M , A where M : A CS , CS is a

L p axy min J P ax, J P ay

gathering of fuzzy subset of S.

~

L N

axy max~

ax, ~

ay a A; x, y V

J N

J N

J N

J N

Definition 2.3. If P, then the bipolar fuzzy soft set over the universe is a pair of (M, A) where

The bipolar fuzzy soft graph is denoted by

~ a.

M : A BF S where BF S is the collection of all bipolar

B ,

Definition 3.2. An bipolar fuzzy soft graph

~ e

~

e , ~ e ,

~ G , ~ , ~ , A

B P , N 1

J P , N

1 L P , N 1

a

a

G J P , N L P , N

is said to be vertex induced if

~ e

~

e , ~ e and

3

3

~

B ,

~

J P , N

~

,

,

L P , N

~

J P , N

a A .

B P , N

~

3 J P , N

e ~

3 L P , N

e ~

e

Definition 3.3. An bipolar fuzzy soft graph

B P , N 5

J P , N

5 , L P , N 5

~ G , ~ , ~ , A is said to be edge induced if

a

a

G

~

B ,

J P , N

~

J P , N

L P , N

,

,

~

L P , N

~

L P , N

a A .

are bipolar fuzzy soft graphs of G .Tabular representation of vertex and edges of bipolar fuzzy soft graphs.

Example3.4.

Consider the bipolar fuzzy graph

G .The parameter set is

denoted by A e ,e ,e , ~

, A and ~ , A

1 3 5 J P , N

L P , N

is a bipolar fuzzy soft set over and E respectively, with

bipolar fuzzy approximation function

~

J P , N

: A BFU

~

and L P , N

: A BFU

~

Fig.3.2.Subgraphs B P , N

e1 ,

~

B P , N

e3 ,

~

B P , N

e5

A/V

e1

e2

e3

e4

e5

e6

e1

(0,0)

(1,-1)

(1,-1)

(0,0)

(1,-1)

(1,-1)

e3

(1,-1)

(1,-1)

(0,0)

(1,-1)

(1,-1)

(0,0)

e5

(1,-1)

(0,0)

(1,-1)

(1,-1)

(0,0)

(1,-1)

A/V

e1

e2

e3

e4

e5

e6

e1

(0,0)

(1,-1)

(1,-1)

(0,0)

(1,-1)

(1,-1)

e3

(1,-1)

(1,-1)

(0,0)

(1,-1)

(1,-1)

(0,0)

e5

(1,-1)

(0,0)

(1,-1)

(1,-1)

(0,0)

(1,-1)

Fig.3.1. Simple Graph G

~

J P , N

a b V / aRb ~

a,b 1,1

d P , N

d P , N

a A

i.e. ~ e e

, e , e , e ,

J P , N 1

2 3 5 6

~

A/E

e1 e2

e2 e3

e3 e4

e4 e5

e5 e6

e6 e1

e1

(0,0)

(1,-1)

(0,0)

(0,0)

(1,-1)

(0,0)

e3

(1,-1)

(0,0)

(0,0)

(1,-1)

(0,0)

(0,0)

e5

(0,0)

(0,0)

(1,-1)

(0,0)

(0,0)

(1,-1)

A/E

e1 e2

e2 e3

e3 e4

e4 e5

e5 e6

e6 e1

e1

(0,0)

(1,-1)

(0,0)

(0,0)

(1,-1)

(0,0)

e3

(1,-1)

(0,0)

(0,0)

(1,-1)

(0,0)

(0,0)

e5

(0,0)

(0,0)

(1,-1)

(0,0)

(0,0)

(1,-1)

J P , N

e3

e1 , e2

, e5

, e6 and

~ e

e , e , e , e

A/E

e1 e3

e3 e5

e5 e1

e1

(0,0)

(1,-1)

(0,0)

e3

(0,0)

(0,0)

(1,-1)

e5

(1,-1)

(0,0)

(0,0)

A/E

e1 e3

e3 e5

e5 e1

e1

(0,0)

(1,-1)

(0,0)

e3

(0,0)

(0,0)

(1,-1)

e5

(1,-1)

(0,0)

(0,0)

J P , N 5 1 3 4 6

~ ~

The edges L P , N a uv E /u, v J P , N a

~

L P , N e1

e2 e3 , e3e5 , e5 e6 ,

~ e

e e , e e

, e e

L P , N 3

2 1 1 5

5 4 and

~ ~ ~

~ e

e e , e e , e e

Definition3.5.Let G1

J P , N , L P , N , A

and

Thus,

L P , N 5

4 3 3 1 1 6

1 1 1 1

~ ~ ~

G2

J P , N , L P , N , B

be the two bipolar fuzzy soft

2 2 2 2

~ ~

graph .Then G2 is the bipolar fuzzy soft subgraph of G1 .

i, B A

ii, B P N xis thesubgraphof B P N x x B

2 ,2 1 ,1

Example3.6.Consider the bipolar fuzzy graph G .The

parameter of two set is denoted by A e1 ,e2 ,e5 and

B e , e , ~ , A and ~ , A is a bipolar

1 2 J P , N L P , N

fuzzy soft set over and E respectively, with bipolar fuzzy

approximation function

~

J P , N

: A BFU

and

~

L P , N

: A BFU

~

Fig.3.4. Subgraphs B P , N

e1 ,

~

B P , N

e2 ,

~

B P , N

e5

Now the bipolar fuzzy approximation function

~

J P , N

: B BFU

~

and L P , N

: B BFU

~

J P , N

a b V / aRb ~

a,b 1,1

d P , N

d P , N

a B

~

Then

e e

, e , e , e ,

J P , N 1

2 3 5 6

~ e

e , e , e

, e

Fig.3.3. Simple Graph G

J P , N 2

1 3 4 6

~ a b V / aRb ~

a, b 1,1 ~

The edges

~

J P , N

a A

d P , N

L P , N

a uv E /u, v J

P , N

aa B

~ ~ e e e , e e , e e

, e e

and

i.e., J

e e , e

, e , e

, e ,

L P , N 1

2 3 3 5 5 6 6 2

P , N 1

1 2 3 5 6 ~

~

J P , N

e2

e1 , e2

, e3 , e4

, e6 and

L P , N e2 e4 e6 , e6 e1 , e1e3 , e3e4 . Hence

~ e

e , e , e , e , e

B A and

J P , N 5

1 3 4 5 6

B P , N

ais the subgraph of

B P , N

a a B

2 2 1 1

J P , N

J P , N

The edges

~

L P , N

a uv E /u, v ~

aa A

~ e e e , e e , e e , e e , e e , e e , e e , e e ,

L P , N 1

1 2 2 3 3 5 5 6

6 2 6 1

5 1 1 3

~ e e e , e e , e e , e e , e e , e e , e e , e e and

L P , N 2 1 2 2 3 3 4 4 6 6 1 1 3 2 4 2 6

~ e e e , e e , e e , e e , e e , e e , e e , e e ~ ~

L P , N 5

1 3 3 4

4 5 5 6

6 1 1 5 5 3 4 6

Fig.5 Subgraphs B P , N e1 , B P , N e2

~ ~ ~

Theorem 3.7. Let G1

J P , N , L P , N , A

and

1 1 1 1

~ ~ ~

G2

J P , N , L P , N , B

be the two bipolar fuzzy soft

2 2 2 2

~ ~

graph .Then G2 is the bipolar fuzzy soft subgraph of G1 if

and only if

~

2 2

2 2

J P , N

a ~

J P , N

J P , N

1 1

aand

~

L P N

L P N

2 ,2

a

~

~

~

J1P ,1N

e if e A \ B

~

L P , N

a

a B .

J P N e J P

N eif e B \ A

~

~

1 1 ,

2 ,2

e

e

~

J P , N

~

J P , N

e if e A B

Proof:

1 1 2 2

Suppose

~

G2 is the bipolar fuzzy soft subgraph of

~

G1 .

~

L1P ,1N

~ e ~

eif e A \ B

eif e B \ A

i, B A

ii,

ii,

~

B P , N

e J P , N

,

,

2

2

~

~

e L P , N

e

L P , N

L P , N

~ 2 2

L1P ,1N

e ~

L P , N

L P , N

2 2

eif e A B

2

2

~

~

2

2

is the subgraph of

~

~

~

~

~

~

B P N e J P

N e

, L P

N e

a B

i.e.,

~ ~ ~ ~

, 1

,

~

~

,

,

1 , 1

G1 E G2 B

P ,

N (e) J

P ,

N e, L P

N e/ e C

~

i.e., J P

N a J P

N a and L P

N a

Example3.9.Consider the bipolar fuzzy graph G .The

~

~

2 ,2

~

1 ,1

2 ,2

parameter of two set is denoted by

L a

a B , since

A e ,e and B e , e , ~

, A

and

1

1

1

1

P , N

2 5 1 4 J P , N

B ais the subgraph of B

a.

~

P , N

P , N

L P , N , A

is a bipolar fuzzy soft set over and E

2 2 1 1

~ ~

respectively, with bipolar fuzzy approximation function

Conversely, Assume that J

2P

,2

N a J

1P

,1

N a and

~

J P , N

: A BFU

and

~

L P , N

: A BFU

~

~

~

~

L P N a L P

N a

a B . Since

~

G is the

2 ,2 1 ,1 1

P , N

P , N

bipolar fuzzy soft graph of G . B a is a bipolar fuzzy

1 1

subgraph of G for all a A.

G2

G2

~

P , N

P , N

Since is the bipolar fuzzy soft graph of G . B a

2 2

is a bipolar fuzzy subgraph of G for all a B .Thus

B P N a is a bipolar fuzzy subgraph of B P N a for

2 , 2

~

1 ,1

all a B .Hence G2 is the bipolar fuzzy soft subgraph of

~

G1 .

Fig.3.6. Simple Graph G

Definition 3.8. Let

G1

J P N , L P , N , A

and

J P N a b V / aRb d P

N a, b 1,1 a A

~ ~ ~

~ ~ ~

1 , 1 1 1

~

,

~

,

2 2 2 2

2 2 2 2

G2

J P , N L P , N , B

be the two bipolar fuzzy soft

~

graphs. Extended union of two bipolar fuzzy soft graphs over

i.e., J P , N e2 e1 , e2 , e3 , e6

and

a common universe U is a bipolar fuzzy soft graph

~ e e , e

, e , e

( B P , N

,C), where C = AB and

B P , N

: C BFU is

J P , N 5

3 4 5 6

J P , N

J P , N

dened by The edges

~

L P , N

a uv E /u, v ~ aa A

~ e e e , e e , e e , e e and

L P , N 2

1 2 2 3 6 1 2 6

~ e

e e , e e , e e , e e

L P , N 5

3 4 4 5 5 6 5 3

B

B

~

P , N

e ~J

~

~

~

~

~

~

P , N

e , ~L

~

~

P , N

e and

~

B P , N

e J P , N

e L P , N

e ,

2

2

1

1

1

1

,

,

1

1

3

3

3

3

~

B P , N

e J P , N

e L P , N

e

~

B P , N

5

5

5

5

,

,

5

5

~

~

~

e J P , N

2

2

~

~

e ~

e L P , N

2

2

,

,

~

~

e , ~

e ,

2

2

e and

Now the bipolar fuzzy approximation function

B P , N 4

J P , N

4 L P , N 4

~

J P , N

: B BFU

and

~

L P , N

: B BFU

B P N e

~

P

P

N e

, ~

P

P

J

J

L

L

N e

~

~

, 5

Therefore

,

5 , 5

J P , N a

b V / aRb d P , N

a, b

(1,1) a B ~

~

G1 E G2 B P , N (e1 ), B P , N (e2 ), B P , N (e4 ), B P , N (e5 )

~

Then

e e

, e and ~

e e , e

~ ~ ~

J P , N 1 2 6

J P , N

J P , N

The edges

J P , N 4 3 5

Theorem 3.10. Let G1

J1P ,1N

, L1P ,1N , A

and

~

L P , N

a uv E /u, v ~

aa B

~ ~

2 2

2 2

G2 J P , N

~

,

,

2 2

2 2

L P , N , B

be the two bipolar fuzzy soft

~ ~

L P , N e1

e6 e2

and L P , N e4

e3e5

graphs with A B and

J

J

B~ e

~J

e , ~L

e and

~

J P N

a ~

P

P

N a for all a A B . Then

4

4

~

~

P , N 1

P , N 1

P , N 1

1 ,1

2 ,2

~

~

~

B P , N

e J P , N

e L P , N

e

their union is the bipolar fuzzy soft graph of G . Proof:

,

,

4

4

4

4

Extended union of two bipolar fuzzy soft graphs

~ ~ ~

~ ~

B

(e) ~

e, ~

e/ e C

Let G1

J P , N , L P , N , A

and

G1 E G2

P , N

J P , N

L P , N

1 1 1 1

~ ~ ~

Where C = AB e1 ,e2 ,e4 ,e5

G2

J P , N , L P , N , B

be the two bipolar fuzzy soft

~ ~ 2 2 2 2

(e)

(e)

e,

e,

J P , N e1 e2 , e6 , L P , N e1 e6 e2

graphs. Extended union of two bipolar fuzzy soft graphs

J

J

~ e

e , e , e , e ,

~ ~

G1 E G2

B

P , N

~

J P , N

~

L P , N

e/ e C

P , N 2

1 2 3 6

~ e

e e , e e , e e , e e

Where C = AB

L P , N 2

1 2 2 3 6 1 2 6

~ e

e , e , ~ e e e ~

J P , N 4

3 5 L P , N 4 3 5

J e if e A \ B

~

1P ,1N

J P , N e5

e3 , e4 , e5 , e6 ,

~ e ~

eif e B \ A

~ e

e e , e e , e e , e e

J P , N J P , N

2 2

2 2

L P , N 5

3 4 4 5 5 6 5 3

~

J P , N

~

e

e

J P , N

e if e A B

1 1 2 2

~

L P , N

~

e

e

L1P ,1N

~

L P , N

eif e A \ B

eif e B \ A

~ 2 2 ~

L

1P ,1

N e L

2 P ,2

N eif e A B

~

Since G1

is the bipolar fuzzy soft graph of G . B

a

Fig.3.7. Subgraphs

P , N

P , N

is a connected bipolar fuzzy subgraph of

1 1

G for all

~ e

, ~

e , ~

e , ~

e

a A \ B . Since

~

is the bipolar fuzzy soft graph of

B P , N 2 B P , N 5

B P , N 1

B P , N 4 G2

Then subgraphs of G

G . B a is a connected bipolar fuzzy subgraph of

N

P

P

2 , 2

J P , N

J P , N

G for all a B \ A .Let a A B and

~ a, ~

~

a J1P ,1N

a ~

2 2

a,

J P , N

L P , N

~

L P , N

~

a

a

L P , N

a

.

1 1 2 2

Since

Since

~

J P , N

~

a ,

a ,

L P , N

aand

Example3.13.Consider the bipolar fuzzy graph

G .The

~ ~ parameter of two set is denoted by

A e1 ,e2 ,e5 and

J P N a, L P N a are connected bipolar fuzzy ~ ~

1

2 ,2

1 1 1

2 ,2

B e1 , e2 , J

P , N

, A

and L

P , N

, A

is a bipolar

subgraph of G by assumption

fuzzy soft set over and E respectively, with bipolar fuzzy

a

a

~

J P , N

~

~

~

J P , N

a for all a A B .

approximation function

~

J P N

: A BFU

and

1

1

~1

2 2

,

~ U

J P , N

a , L P , N a

is a connected bipolar fuzzy

L P , N

: A BF

(e)

(e)

subgraph of G and

e,

e,

~ ~

G1 E G2

B

P , N

~

J P , N

~

L P , N

e/ e C

is a bipolar soft fuzzy graph of G .

~ ~ ~

Definition 3.11. Let

G1

J P , N , L P , N , A

and

~ ~ ~

1 1 1 1

G2

J P , N , L P , N , B

be the two bipolar fuzzy soft

2 2 2 2

graphs. Restricted union of two bipolar fuzzy soft graphs over a common universe U is a bipolar fuzzy soft graph

( B P , N ,C), where C = dened by

A B and B P , N

: C BFU is

~ ~ Fig.3.8. Simple Graph G

N a ~

N a ~

,

,

~ ~ J P , N a J P , N a, ~ ~

J P

N a, L

1

P , L

1 2

L

L

a ~

2

a

J P , N a b V / aRb d P , N a, b rad G a A

1P ,1N

2P ,2 N

i.e.,

~

e e

, e , e , e ,J P , N 1

2 3 5 6

Definition 3.12. Let

~ ~ , ~

, A and

~ e

e , e , e , e and

G1 J P , N L P , N

J P , N 2

1 3 4 6

~ ~ ~

1 1 1 1

~ e

e , e , e , e

G2

J 2 P ,2

N , L

2 P ,2

N , B

be the two bipolar fuzzy soft

J P , N 5

1 3 4 6

graphs. Extended intersection of two bipolar fuzzy soft

The edges

graphs over a common universe U is a bipolar fuzzy soft ~

a uv E /u, v ~

aa A

graph ( B

P , N

,C), where C =

A Band

L P , N

~

J P , N

B : C BFU

is dened by

L P , N e1

e2 e3 , e3e5 , e5 e6 , e6 e2 ,

P , N

~

L P , N e2

e3e4 , e4 e6 , e6 e1 , e1e3

and

~ ~ e

e e , e e , e e , e e

J

J

1P ,1N

e if e A \ B

L P , N 5

1 3 3 4

6 1 4 6

e

e

~

J P , N

~

J P , N

eif e B \ A

~ 2 2 ~

J

1P ,1

N e J

2 P ,2

N e if e A B

~

L P , N

~

e

e

L1P ,1N

~

L P , N

eif e A \ B

eif e B \ A

i.e.,

(e)

(e)

~ 2 2 ~

L

1P ,1

N e L

2 P ,2

N eif e A B

~ ~

G1 E G2

B

P , N

~

J P , N

~

e,

e,

L P , N

e/ e C

Fig.3.9. Subgraphs

~

B P , N

e1 ,

~

B P , N

e2

~

,

,

B P , N

e5

~

Fig.3.10. Subgraphs

e , ~

e

Now the bipolar fuzzy approximation function

B P , N 1

B P , N 2

~

J P , N

: B BFU

~

and L P , N

: B BFU

Then subgraphs of G

~

~

~

~

~

J P , N

a b V / aRb ~

a,b rad G

~

B P , N

~

B P , N

e J P , N

d P , N

d P , N

1

1

2

2

5

5

~

~

~

~

e J P , N

e L P , N

1

1

2

2

,

,

5

5

,

,

~

~

~

~

e L P , N

e ,

,

,

1

1

2

2

5

5

e and

a B

Then

~

~

B P , N

Therefore

e J P , N

e L P , N

e

J P , N e1 e1 , e2 , e3 , e5 , e6 ,

~ ~

G1 E G2

B

P , N

(e1 ), B

P , N

(e2

), B

P , N

(e5 )

~

J P , N

e2

e1 , e2

, e3 , e4

, e6

~ ~ ~

The edges

Definition 3.14. Let G1 J P , N , L P , N , A

and

~

~

1 1 1 1

L P , N a

uv E / u, v

J P , N a a B

~ ~ ~

G2 J P , N , L P , N , B

be the two bipolar fuzzy soft

~ e e e

, e e , e e , e e

, e e

, e e , e e , e e 2 2 2 2

L P , N 1

1 2 2 3 3 5 5 6 6 2

6 1 5 1 1 3

graphs. Restricted intersection of two bipolar fuzzy soft

and

~

e e e

, e e , e e , e e

, e e , e e

, e e

, e e

graphs over a common universe U is such that

A B

L P , N 2

1 2 2 3

4 3 4 6

6 1 2 4

2 6 1 3

bipolar fuzzy soft graph ( B P , N ,C), where C =

A B and

(e)

(e)

The extended intersection of two bipolar soft fuzzy graph is

B P , N

: C BFU

is dened by

e,

e,

~ ~

G1 E G2

B

P , N

~

J P , N

~

L P , N

e/ e C

where C =

~

A B

~ a, ~

~

a J1P ,1N

a ~

J P , N

J P , N

2 2

a,

J P , N e1 e1 , e2 , e3 , e5 , e6 ,

J P , N

L P , N

~

a ~ a

~

L1P ,1N

L P , N

2 2

2 2

L P , N e1

e1e2 , e2 e3 , e3e5 , e5 e6 , e6 e2 , e6 e1 , e5 e1 , e1e3

~

J P , N

e2

e1 , e2

, e3 , e4

, e6 ,

REFERENCES

  1. A. Rosenfeld, Fuzzy graphs, in: L.A. Zadeh, K.S. Fu, K.

    ~ e

    e e , e e , e e , e e , e e , e e

    , e e , e e

    Tanaka and M. Shimura,(editors), Fuzzy sets and its application

    L P , N 2

    and

    1 2 2 3

    4 3 4 6

    6 1 2 4

    2 6 1 3

    to cognitive and decision process, Academic press, New York (1975) pp. 77 95.

  2. R.T.Yeh and S.Y.Bang, Fuzzy relations, fuzzy graphs and their applications to clustering analysis, in: L.A. Zadeh, K.S. Fu, K.

    e

    e

    ~

    J P , N 5

    e1 , e3 , e4 , e6 ,

    Tanaka and M. Shimura,(editors), Fuzzy sets and its application to cognitive and decision process, Academic press, New York (1975) 125 149.

    e

    e

    ~

    L P , N 5

    e1e3 , e3e4

    , e6 e1 , e4 e6 Finally, complete

  3. A. Aygünoglu, H. Aygün, Introduction to fuzzy soft groups, Computers and Mathematics with Applications 58(2009), 1279- 1286.

  4. P.K.Maji, A.R.Roy, R.Biswas, On intuitionistic fuzzy soft sets, The journal of fuzzy mathematics 12(2004), 669-683.

  5. Saleem Abdullah, Muhammad Aslam and Kifayat Ullah, Bipolar Fuzzy Soft sets and its applications in decision making problem,

    Journal of Intelligent and Fuzzy, March 2014

  6. T. Pathinathan and J. Jesintha Rosline 2014 Matrix Representation of Double layered fuzzy graph and its properties, Annals of Pure and Applied Mathematics, Vol. 8, No. 2, pp. 51 58

  7. Muhammad Akram, Saira Nawaz, on fuzzy soft graphs, Italian journal of pure and applied mathematics, 34(2015), 497-514.

  8. Xu, W., Ma, J., Wang, S., Hao, G., Vague soft sets and their properties, Computers and Mathematics with Applications 59(2010), 787-794.

  9. M.Rajeshwari, R. Murugesan, K. A. Venkatesh, Properties of Bipolar fuzzy soft graphs, International Journal of Mechanical Engineering and Technology 9(13) 2018,pp.537-546.

  10. A.Nagoorgani and K.Radha, The degree of a vertex in some fuzzy graphs, Inter. Journal of Algorithms, Computing and Mathematics, 2(3) (2009) 107 – 116.

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