Operations on Dombi Bipolar Fuzzy Graphs Using T-Operator

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Operations on Dombi Bipolar Fuzzy Graphs Using

T-Operator

R. Jahir Hussain and S. Satham Hussain

PG and Research Department of Mathematics

Jamal Mohamed College, Trichy - 620 020, Tamilnadu, India.

Abstract—In this manuscript, the dombi bipolar fuzzy graphs are introduced. The several operations like union, ring sum, direct product and Cartesian product of Dombi bipolar fuzzy graphs are established and the proposed concepts are described with the suitable examples.

Index Terms—Bipolar fuzzy graphs; Cartesian product; Dombi fuzzy graphs; Operations.

The bipolar fuzzy models give more accurate precision, flexibility and compatibility to the system when compared to the classical and fuzzy models. In 1994, Zhang [11], [12] introduced the concept of bipolar fuzzy sets as a generalization of fuzzy sets. Well known that, bipolar fuzzy sets are an extension of fuzzy sets whose membership degree range is [−1,1]. In a bipolar fuzzy set, the membership degree of an element means that the element is irrelevant to the corresponding property, the membership degree (0,1] of an element indicates that the element somewhat satisfies the property, and the membership degree [−1,0) of an element indicates that the element which satisfies the implicit counter property. In many fields, it is crucial to deal with bipolar information. It is stressed that positive information represents what is granted to be possible, where as negative information represents what is considered to be impossible. This domain has recently motivated in literature in several directions. For instance, consider the spatial relations: Human beings consider

left andright as opposite directions. But this does not mean that one of them is the negation of the other. The semantics of oppositecaptures a notion of symmetry rather than a strict complementation. In particular, there may be positions which are considered neither to the right nor to the left of some reference object, thus leaving some room for indetermination. This corresponds to the idea that the union of positive and negative information does not cover the whole space. In 2011, Akram [1] introduced the notion of bipolar fuzzy graphs with various methods of their construction and also discussed the concept of isomorphism of these graphs. The bipolar fuzzy influence graph of a social group is described in [2]. Bipolar regular graphs and a result on an isomorphism between two bipolar fuzzy graphs follows from an isomorphism of their corresponding bipolar fuzzy line graphs are investigated in [3]. Recently, bipolar fuzzy graphs received much attention like bipolar fuzzy competition graphs are applied in politics see [7], [9], [10] and references therein. In [6], using the concept

of path and strength of connectedness of bipolar fuzzy graphs, triple connected domination bipolar fuzzy graph is established. New operations on bipolar fuzzy graph namely direct product, semi strong product and strong product are studied in [8].

The operations of t-norm and t-conorm introduced by Dombi [5] is said to be Dombi operations which leads to good flexibility with the general parameter. The Dombi operations have so far not yet been applied for bipolar fuzzy graphs. In this paper, we defined some operations on Dombi bipolar fuzzy graphs. The goal of this paper is to mention that the

maxandmin operations are not the only candidates for the generalization of the classical graphs to fuzzy graphs. We propose to incorporate the advancement proposed in [4], here we employed an operator, particularly T-operator, namely the Dombi operator in the area of bipolar fuzzy graph theory and the proposed concepts are provided with an appropriate examples.

In this paper, the concept of Dombi bipolar fuzzy graph is introduced and obtained that the direct product, join and ring sum of two Dombi bipolar fuzzy graphs are the Dombi bipolar fuzzy graphs. Generally, the Cartesian product and strong product of two Dombi bipolar fuzzy graphs are not Dombi bipolar fuzzy graphs. However, these graphs may be Dombi bipolar fuzzy if they have crisp vertices. Moreover, these results on Dombi bipolar fuzzy graphs are considered preserving strong property. In the real world, the Dombi bipolar fuzzy graph can be a description of the uncertainty of all kinds of networks. The theoretical results presented in this paper will improve the results mentioned in [4], and in future work, we investigate metric aspects of neutrosophic soft Dombi bipolar fuzzy graphs and the degree of the resultant graphs will also determined.

I. PRELIMINARIES

The following definitions are used to prove the main results.

Definition 1.1:[4] A binary operation T : [0,1]×[0,1]→ [0,1]is a triangular norm (t-norm), if for everyu, v, w∈[0,1],

which satisfies the following conditions: (1) T(1, u) =u

(2) T(u, v) =T(v, u)

International Journal of Research in Advent Technology, Vol.7, No.5, May 2019

E-ISSN: 2321-9637

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(3) T(u, T(v, w)) =T(T(u, v), w)

(4) T(u, v)≤T(u, w)ifv≤w.

Definition 1.2: Possible choices for norms and dual t-conorms are as follows:

♠The minimum operatorM : M(u, v) = min(u, v) ♠The maximum operator M∗: M∗(u, v) = max(u, v) ♠The product operator P:P(u, v) =uv

♠The probabilistic operator P∗: P∗(u, v) =u+v−uv ♠The Lukasiewicz’s t-normW :W(u, v) = max(u+v−1,0) ♠The bounded sum W∗: W∗(u, v) = min(u+v,1). We define the set of T−operator as

T(u, v) = uv u+v−uv

which is obtained by taking λ= 0 in Hamacher family and

λ= 1in Dombi family of t-norms and t-conorms. Also it fol-lows that,P(u, v)≤uv/(u+v−uv)≤M(u, v), M∗(u, v)≤ uv/(u+v−uv)≤P∗(u, v).

Definition 1.3:LetXbe a non-empty set. Then the mapping

A= (µP A, µ

N

A) :X×X→[0,1]×[−1,0]a bipolar fuzzy

re-lation onX such thatµP

A(xy)∈[0,1]andµNA(xy)∈[−1,0]. Definition 1.4:Let A= (µP

A, µ N

A)andB= (µ P B, µ

N B)be a

bipolar set onX.IfA= (µP A, µ

N

A)is a bipolar fuzzy relation

on a set X thenA= (µP A, µ

N

A) is said to be a bipolar fuzzy

relation onB = (µP

B, µNB).IfµPB(x, y)≤min(µPB(x), µPB(y))

and µN

B(x, y)≥max(µNB(x), µNB(y)) for every x, y∈X. A

bipolar fuzzy relation A on X is symmetric if µP

A(x, y) = µP

A(y, x)andµNA(x, y) =µNA(y, x)for every X.

II. DOMBIBIPOLARFUZZYGRAPHS

Definition 2.1:A dombi bipolar fuzzy graphG= (V, A, B)

is a non-empty finite set onV together with a pair of functions

A = (µP A, µ

N

A) : V → [0,1]×[−1,0]and B = (µ P B, µ

N B) : V ×V →[0,1]×[−1,0]such that for everyx, y∈V,

B₁P(xy)≤ A P

1(x)AP1(y)

AP

1(x) +AP1(y)−AP1(x)AP1(y)

and

B₁N(xy)≥ A N

1(x)A

N

1(y)

AN

1 (x) +AN1 (y)−AN1(x)AN1 (y)

Then one can say that A = (AP

1, AN1) as the dombi bipolar

vertex set ofGandB= (BP

1, BN1 )as the dombi bipolar edge

set of G.

Example 2.1:Consider a dombi bipolar fuzzy graphGsuch that A= (a, b, c, d), B ={(ab),(bc),(cd),(da)}. By routine evaluation, we have

G

F igure1 Dombi Bipolar Fuzzy Graph

Definition 2.2:LetAibe the bipolar fuzzy subset ofViand letBibe the fuzzy subset ofEi, i= 1,2, . . . .Define the direct product of (G1×G2) = (A1×A2, B1×B2) of the dombi

bipolar fuzzy graphs G1 = (A1, B1) and G2 = (A2, B2)

respectively, given as follows: (i) (AP

1 ×AP2)(x1, x2) =

AP

1(x1)AP2(x2)

AP

1(x1)+AP2(x2)−AP1(x1)AP2(x2) for

every(x1, x2)∈V1×V2,and

(AN

1 × AN2)(x1, x2) =

AN₁(x1)AN2(x2)

AN

1(x1)+AN2(x2)−AN1(x1)AN2(x2) for

every(x1, x2)∈V1×V2

(ii)

(B₁P×B₂P)((x1x2),(y1, y2))

= B

P

1(x1y1)B2P(x2y2)

BP

1(x1y1) +BP2(x2y2)−BP1(x1y1)BP2(x2y2)

for every(x1x2, y1y2)∈E1×E2and

(BN₁ ×B₂N)((x1x2),(y1, y2))

= B

N

1 (x1y1)B2N(x2y2)

BN

1 (x1y1) +B2N(x2y2)−BN1 (x1y1)B2N(x2y2)

for every(x1x2, y1y2)∈E1×E2

Example 2.2: Consider two dombi bipolar

fuzzy graph G1 and G2, where G1 = (A1, B1)

G1 = {a(0.5,−0.6), b(0.4,−0.3)}, G2 =

{c(0.7,−0,4), d(0.5,−0.4), e(0.6,−0.5)}

(G1×G2)((ac)(bd)) =(0.14,−0.06)

(G1×G2)((ae)(bd)) =(0.13,−0.12)

(G1×G2)((ad)(bc)) =(0.14,−0.06)

(G1×G2)((ad)(be)) =(0.13,−0.062)

Hence it is easy to verify that G1×G2 is the dombi bipolar

graph ofG1 andG2.

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E-ISSN: 2321-9637

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F igure2 Direct product of Dombi Bipolar Fuzzy graph

Proposition 2.3:LetG1andG2 be the dombi bipolar fuzzy

graphs then the direct product G1×G2 of G1 andG2 is the

dombi bipolar fuzzy graph.

Proof: Considerx1y1∈E1, x2y2∈E2.Then

(B₁P×BP₂)((x1x2),(y1y2)) =T(B1P(x1y1), BP2(x2y2))

≤ A

P

1(x1)AP1(y1)

AP

1(x1) +AP1(y1)−AP1(x1)AP1(y1)

,

AP

2(x2)AP2(y2)

AP

2(x2) +AP2(y2)−AP2(x2)AP2(y2)

By putting a=η1(x1), b=η1(y1), c=η2(x2), d=η2(y2),

we get

(BP₁ ×B₂P)((x1x2),(y1y2))≤T

ab a+b−ab,

cd c+d−cd

=

abcd

(a+b−ab)(c+d−cd)

ab a+b−ab+

cd c+d−cd−

abcd

=

acbd

(a+c−ac)(b+d−bd)

ac a+c−ac+

bd b+d−bd −

acbd

=[(A1×A2)((x1, x2))(A1×A2)((y1, y2))]

.

[(A1×A2)P((x1, x2)) + (A1×A2)P((y1, y2))

−(A1×A2)P((x1, x2))(A1×A2)P((y1, y2))]

Similarly

(BN₁ ×B₂N)((x1x2),(y1y2))≤T

ab a+b−ab,

cd c+d−cd

=

abcd

ab a+b−ab+

cd c+d−cd+

abcd

=

acbd

ac a+c−ac+

bd b+d−bd+

acbd

=[(A1×A2)N((x1, x2))(A1×A2)N((y1, y2))]

.

[(A1×A2)N((x1, x2)) + (A1×A2)N((y1, y2))

−(A1×A2)N((x1, x2))(A1×A2)N((y1, y2))]

Hence the direct productG1×G2is the dombi bipolar fuzzy

graph.

Definition 2.4:LetAibe a bipolar fuzzy subset ofViand let Bibe a fuzzy subset ofEi, i= 1,2, . . . .Define the Cartesian

productG1G2= (A1A2, B1B2)of dombi bipolar fuzzy

graph G1 = (A1, B2) and G2 = (A2, B2) of (V1, E1) and

(V2, E2), respectively,

(i)(A1A2)P(x1x2) =

AP

1(x1)AP2(x2)

AP

1(x1)+AP2(x2)−AP1(x1)AP2(x2)

for all(x1, x2)∈V1×V2

(ii) (A1A2)N(x1x2) =

AN

1(x1)AN2(x2)

AN

1(x1)+AN2(x2)−AN1(x1)AN2(x2)

for all(x1, x2)∈V1×V2

(iii) (B1B2)P((xx2)(xy2)) =

AP₁(x)BP₂(x2y2)

AP

1(x)+BP2(x2y2)−AP1(x)BP2(x2y2) for allx∈V1, x2y2∈E2

(iv) (B1B2)N((xx2)(xy2)) =

AN

1(x)BN2(x2y2)

AN

1(x)+BN2(x2y2)−AN1(x)BN2(x2y2) for allx∈V1, x2y2∈E2

(v) (B1B2)P((x1z)(y1z)) =

AP₂(z)BP₁(x1y1)

AP

2(z)+B1P(x1y1)−AP2(z)B1P(x1y1) for allx1, y1∈E1z∈E2

(vi) (B1B2)N((x1z)(y1z)) =

AN

2(z)BN1(x1y1)

AN

2(z)+B1N(x1y1)−AN2(z)B1N(x1y1) for allz∈V2 x1y1∈E1

Remark 2.5: The Cartesian product of two bipolar dombi bipolar fuzzy graphs is not necessarily a dombi bipolar fuzzy graph.

Example 2.3: Consider two dombi fuzzy graphs G1G2.

Then we have

(B1B2)((ac),(ad)) = (0.25,−0.11)

(B1B2)((ad),(ae)) = (0.20,−0.09)

(B1B2)((ae),(be)) = (0.2,−0.046)

(B1B2)((be),(bd)) = (0.2,−0.13)

(B1B2)((bd),(bc)) = (0.18,−0.13)

(B1B2)((ad),(bd)) = (0.2,−0.05)

(B1B2)((ac),(bc)) = (0.2,−0.046).

ClearlyG1G2 is not a dombi bipolar fuzzy graphs.

Definition 2.6:Let Ai be a bipolar fuzzy subset ofVi and

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F igure3 Cartesian product of dombi bipolar fuzzy graph

G1∪G2= (A1∪A2, B1∪B2)of dombi bipolar fuzzy graphs

G1= (A1, B1)andG2= (A2, B2)be given as follows:

(A1∪A2)P(x) =   

 

AP

1(x) if x∈V1\V2,

AP2(x) if x∈V2\V1, AP₁(x)AP₁(y)

AP

1(x)+AP1(y)−AP1(x)AP1(y)

ifx∈V1∩V2.

(A1∪A2)N(x) =   

 

AN

1 (x) if x∈V1\V2,

AN2 (x) if x∈V2\V1, AN₁(x)AN₁(y)

AN

1(x)+AN1(y)−AN1(x)AN1(y)

ifx∈V1∩V2.

Also,

(B1∪B2)P(xy) =     

   

BP

1(xy)/2 if xy∈E1\E2,

BP2(xy)/2 if xy∈E2\E1, AP

1(x)AP1(y)

AP

1(x)+AP1(y)−AP1(x)AP1(y)

. 2

ifxy∈E1∩E2.

(B1∪B2)N(xy) =     

   

BN

1 (xy)/2 if xy∈E1\E2,

BN2 (xy)/2 if xy∈E2\E1, AN

1(x)AN1(y)

AN

1(x)+AN1(y)−AN1(x)AN1(y)

. 2

ifxy∈E1∩E2.

Example 2.4: The dombi fuzzy graphs are,

G1 = {j(0.5,−0.5), k(0.8,−0.8), l(0.4,−0.4)} and

G2 = {j(0.6,−0.6), k(0.5,−0.5), m(0.3,−0.3)}. Then we

have A1∪A2 = ₍₀_.₃₇_,j₋₀_.₂₁₎,₍₀_.₄_,₋k₀_.₂₃₎,₍₀_.₄_,₋l₀_.₄₎,₍₀_.₃_,m₋₀_.₃₎

B1∪B2= ₍₀_.₀₅_,jl₋₀_.₀₉₎,₍₀_.₁₁_,jk₋₀_.₀₉₎,₍₀_.₀₅mk_,₋₀_.₀₇₅₎,₍₀_.₁₅lk_,₋₀_.₁₎

F igure4 Union of dombi bipolar fuzzy graph

Theorem 2.7:The unionG1∪G2ofG1andG2is the dombi

bipolar fuzzy graph of G if and only if G1 and G2 are the

dombi bipolar fuzzy graphs of G, where A1, A2, B1 andB2

are bipolar subsets ofV1, V2, E1andE2respectively, provided

V1∩V2=φ.

Proof:Assume thatG2∪G2 are dombi bipolar fuzzy graphs.

Letxy∈E1,thenxy∈E2, andx, y∈V1\V2.Then

BP₁(xy) = (B1∪B2)P(xy)

≤[(AP₁ ∪A₂P)(x)(AP₁ ∪AP₂)(y)].[(AP₁ ∪AP₂)(x)

+ (AP₁ ∪AP₂)(y)−(AP₁ ∪AP₂)(x)(AP₁ ∪AP₂)(y)]

BP₁(xy)≤ A P

1(x)AP1(y)

AP

ThusG1 is the dombi bipolar fuzzy graph of G.Similarly it

is easy to verify that G2 is the dombi bipolar fuzzy graph of

G. Conversely, assume thatG1 andG2 are the dombi bipolar

fuzzy graph ofGrespectively, considerxy∈E1\E2then by

the definition of union, it follows that,

(B1∪B2)P(xy) =B1P(xy)≤(A

P

1(x), A

P

1(y))

=((A1∪A2)P(x),(A1∪A2)P(x)).

In the similar way, we can findxy∈E2\E1,

(B1∪B2)P(xy)≤((A1∪A2)(x),(A1∪A2)(x))

≤[(A1∪A2)P(x)(A1∪A2)P(y)]

.

[(A1∪A2)P(x)

+ (A1∪A2)P(y)−(A1∪A2)P(x)(A1∪A2)P(y)].

Also,

B1N(xy)≤

AN

1(x)AN1(y)

AN

1 (x) +AN1 (y)−AN1(x)AN1 (y)

(B1∪B2)N ≤[(A1∪A2)N(x)(A1∪A2)N(y)]

.

[(A1∪A2)N(x)

+ (A1∪A2)N(y)−(A1∪A2)N(x)(A1∪A2)N(y)].

Definition 2.8:Let Ai be a bipolar fuzzy subset ofVi and letBibe a bipolar fuzzy subset ofEi, i= 1,2. . . .Define the ring sumG1⊕G2= (A1⊕A2, B1⊕B2)of the dombi bipolar

fuzzy graphs, where G1 = (A1, B1) and G2 = (A2, B2)

satisfies the following conditions: (i)

(A1⊕A2)P(x) = (A1∪A2)P(x) if x∈V1∪V2

(ii)

(A1⊕A2)N(x) = (A1∪A2)N(x) if x∈V1∪V2

(iii)

(B1⊕B2)P(xy) =

  

BP

1(xy), if xy∈E1\E2,

BP

2(xy), if xy∈E2\E1,

0, if xy∈E1∩E2.

(iv)

(B1⊕B2)N(xy) =

  

BN

1 (xy), if xy∈E1\E2,

BN

2 (xy), if xy∈E2\E1,

0, if xy∈E1∩E2.

Proposition 2.9:The ring sumG1⊕G2of two bipolar dombi

fuzzy graphs G1 and G2 of G∗1and G∗2 is the dombi bipolar

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Proof:Considerxy∈E1\E2,then there are three possibilities

such as:

(i) x, y∈V1\V2,

(ii) x∈V1\V2, y∈V1∩V2,

(iii) x, y∈V1∩V2.

(i)Supposex, y∈V1\V2,then by the Definition of ring sum

we have,

(B1⊕B2)P(xy) =B1P(xy)

≤(AP₁(x), AP₁(y))

≤((A1∪A2)P(x),(A1∪A2)P(y))

≤((A1⊕A2)P(x),(A1⊕A2)P(y))

(ii)Supposex∈V1\V2, y∈V1∩V2,

(B1⊕B2)P(xy) =BP1(xy)

≤(AP₁(x), AP₁(y))

≤((A1∪A2)P(x), AP1(y))

≤((A1⊕A2)P(x), AP1(y))

Clearly,

A1(y)≤

A1(y) +A2(y)−2A1(y)A2(y)

1−A1(y)A2(y)

.

By substituting, A1(y) =f, A2(y) =h,we have

f ≤f+h−2f h 1−f h f−f2h≤f+h−2f h

0≤(f−1)2.

Since (B1⊕B2)(xy)≤((A1⊕A2)(x),(A1⊕A2)(y))

(iii)Supposex, y∈V1∩V2. Then we get,

(B1⊕B2)(xy) =B1(xy)

≤(A1(x), A1(y))

≤((A1∪A2)(x),(A1∪A2)(y))

≤((A1⊕A2)P(x),(A1⊕A2)(y))

Since by (ii), we have

A1(x)≤

A1(x) +A2(x)−2A1(x)A2(x)

1−A1(x)A2(x)

and

A1(y)≤

A1(y) +A2(y)−2A1(y)A2(y)

1−A1(y)A2(y)

Because of symmetry, we obtain forxy∈E2\E1in the three

possible cases.

(B1⊕B2)P(xy)≤((A1⊕A2)P(x),(A1⊕A2)P(y))

≤[(A1⊕A2)P(x)(A1⊕A2)P(y)]

.

[(A1⊕A2)P(x)

+ (A1⊕A2)P(y)−(A1⊕A2)P(x)(A1⊕A2)P(y)]

Hence proved.

Definition 2.10:The complement of D-Bipolar-fuzzy graphs

G = (A, B) of G = (V, E) is a D-Bipolar-fuzzy graphs of

F igure5Complement of dombi bipolar fuzzy graph

G= (A, B)whereA(x) =Afor all x∈V andB is defined as follows.

BP(xy) =     

AP₍_x₎_AP₍_y₎

AP₍_x₎₊_AP₍_y₎₋_AP₍_x₎_AP₍_y₎, ifB

P₍_xy_{) = 0}

AP₍_x₎₊_AP₍_y₎₋_AP₍_x₎_AP₍_y₎−A

P₍_xy₎_,

if0≤BP(xy)≤1

BN(xy) =     

AN(x)AN(y)

AN₍_x₎₊_AN₍_y₎₋_AN₍_x₎_AN₍_y₎, if B

N₍_xy_{) = 0}

AN₍_x₎_AN₍_y₎

AN₍_x₎₊_AN₍_y₎₋_AN₍_x₎_AN₍_y₎−A

N₍_xy₎_,

if −1≤BN₍_xy₎_≤₀

Example 2.5: Consider a D-Bipolar-fuzzy graph overV = {j, k, l, m}defined by

A=n j

0.3,−0.3, k 0.6,−0.6,

l 0.4,−0.4,

m 0.9,−0.9

o ,

B =n jm

0.29,−0.18, km 0.56,0.26,

lm 0.2,−0.1

o

we have

BP₍_xy_{) =} A

P₍_x₎_AP₍_y₎

AP₍_x_{) +}_AP₍_y_{) +}_AP₍_x₎_AP₍_y₎

−BP₍_xy₎

= A

AP₍_x_{) +}_AP₍_y₎₋_AP₍_x₎_AP₍_y₎

− A

AP₍_x_{) +}_AP₍_y₎₋_AP₍_x₎_AP₍_y₎−B P₍_xy₎

=BP(xy)for every x, y∈V.

HenceG=G

Definition 2.11:A homomorphism χ: (G1→G2)of two

D-Bipolar-fuzzy graphs G1 = (A1, B1) and G2 = (A2, B2)

is a mapping χ:V1→V2 satisfying the following condition:

(i) AP₁(x)≤A₂P(χ(x))for allx∈V1

(ii) AN1 (x)≥A2N(χ(x))for allx∈V1

(ii) B1P(xy)≤B2P(χ(xy))for allxy∈V2

(iv) BN

1 (xy)≥BN2 (χ(xy))for allxy∈V2

Definition 2.12: A isomorphism χ : (G1 → G2) of

two D-Bipolar-fuzzy graphs G1 = (A1, B1) and G2 =

(A2, B2)(denoted as G1 ∼= G2) is a bijective mapping χ :

V1→V2 satisfying the following condition:

(v) AP

1(x) =AP2(χ(x))for allx∈V1

(vi) AN

(6)

(vii) BP

1(xy) =B2P(χ(xy))for allxy∈V2

(viii) BN

1 (xy) =B2N(χ(xy))for allxy∈V2

A weak isomorphismχ: (G1→G2)is a bijective

homomor-phism with the condition (v) and (vi) above and a co-weak isomorphism χ : (G1 → G2) is a bijective homomorphism

with the condition (vii) and (viii) above

Definition 2.13: A D-Bipolar-fuzzy graph G = (A, B) is said to be Self-complementary ifG= (A, B)∼=G= (A, B). Proposition 2.14:LetG= (A, B)be a self complementary D-Bipolar-fuzzy graph, then

P x6=y

BP₍_xy_{) =} 1 2

P x6=y

AP₍_x₎₊_AP₍_y₎₋_AP₍_x₎_AP₍_y₎ and

P x6=y

BN₍_xy_{) =}1 2

P x6=y

AN₍_x₎₊_AN₍_y₎₋_AN₍_x₎_AN₍_y₎

Proof:LetGbe a self-complementary D-Bipolar-fuzzy graph. then there exists an isomorphism χ : V → V such that

AP₍_χ₍_x_{)) =} _AP₍_x₎ _{for all} _x _∈ _V _and _AP₍_χ₍_x₎_χ₍_y_{)) =} AP₍_xy₎_{for all}_xy_∈_E. _{By definition of}_G_{, we have}

BP₍_χ₍_x₎_χ₍_y_{)) =} A

P_χ₍_x₎_AP_χ₍_y₎

AP_χ₍_x_{) +}_AP_χ₍_y_{) +}_AP_χ₍_x₎_AP_χ₍_y₎ −BP(χ(x)χ(y))

BP(xy) = A

AP₍_x_{) +}_AP₍_y₎₋_AP₍_x₎_AP₍_y₎ −BP(χ(x)χ(y))

X x6=y

BP(xy) = A

AP₍_x_{) +}_AP₍_y₎₋_AP₍_x₎_AP₍_y₎

−BP(χ(x)χ(y)) X

x6=y

BP(xy) +X x6=y

BP(χ(x)χ(y))

=X

x6=y

AP(x)AP(y)

AP₍_x_{) +}_AP₍_y₎₋_AP₍_x₎_AP₍_y₎

2X

x6=y

BP(xy) =X x6=y

AP₍_x_{) +}_AP₍_y₎₋_AP₍_x₎_AP₍_y₎

X x6=y

BP(xy) =1 2

X x6=y

AP₍_x_{) +}_AP₍_y₎₋_AP₍_x₎_AP₍_y₎

Similarly we prove that

P x6=y

BN₍_xy_{) =} 1 2

P x6=y

AN(x)AN(y)

AN₍_x₎₊_AN₍_y₎₋_AN₍_x₎_AN₍_y₎. Hence

proved.

Proposition 2.15: Let G= (S, A) be the D-Bipolar-fuzzy graph ofG∗.

If BP(xy) = 1₂_AP₍_x₎₊A_APP₍(x_y₎)A₋P_A(Py₍)_x₎_AP₍_y₎ and B

N₍_xy_{) =}

1 2

AN₍_x₎₊_AN₍_y₎₋_AN₍_x₎_AN₍_y₎ for all x, y∈ V, then Gis

self-complementary.

Proof: Let G be the D-Bipolar-fuzzy graph satisfying

BP₍_xy_{) =} 1 2

AP₍_x₎₊_AP₍_y₎₋_AP₍_x₎_AP₍_y₎ for all x, y ∈V,then

the identity mappingI:V →V is an isomorphism fromGto

G. Clearly,I satisfies the condition(v)and(vi)of definition 3.13. Since BP(xy) =1₂_AP₍_x₎₊A_APP₍(_yx₎)A₋P_AP(y₍)_x₎_AP₍_y₎,we have

BP₍_I₍_x₎_I₍_y_{)) =}_B₍_xy₎

= A

AP₍_x_{) +}_AP₍_y₎₋_AP₍_x₎_AP₍_y₎−B P₍_xy₎

= A

AP₍_x_{) +}_AP₍_y₎₋_AP₍_x₎_AP₍_y₎

−1 2(

AP(x)AP(y)

AP₍_x_{) +}_AP₍_y₎₋_AP₍_x₎_S₍_y₎)

= 1 2(

AP₍_x_{) +}_AP₍_y₎₋_AP₍_x₎_AP₍_y₎) =B P₍_xy₎

Thus the condition (vii) and (viii) of Definition 3.13 is also satisfied byI thereforeGis self complementary.

Proposition 2.16:The complements of two isomorphic D-Bipolar-fuzzy graphs are isomorphic and conversely.

Proof:Suppose thatG1andG2are two isomorphic

D-Bipolar-graphs. then there exists a bijective mapping χ : V1 → V2

satisfying

(i) AP₁(x) =AP₂(χ(x))for allx∈V1

(ii) AN1 (x) =AN2(χ(x))for allx∈V1

(iii) B1P(xy) =B2P(χ(xy))for allxy∈V2

(iv) B1N(xy) =BN2 (χ(xy))for allxy∈V2

using the definition of complement, we have

BP

1(xy) =

AP

1(x)AP1(y)

AP

−B₁P(xy)

= S2(χ(x))A P

2(χ(y))

AP

2(χ(x)) + (AP2χ(y))−(AP2χ(x))(AP2χ(y))

−B2P(χ(x)χ(y)) =B2P(χ(x)χ(y)).

henceG1∼=G2. Similarly, we can prove the converse part.

III. STRONGD-BIPOLAR-FUZZY GRAPH

Definition 3.1:A strong D-Bipolar-fuzzy graphG= (A, B)

is called a strong D-N-fuzzy graph ofG= (V, E)if

BP(xy) = A

AP₍_x_{) +}_AP₍_y₎₋_AP₍_x₎_AP₍_y₎

and

BN(xy) = A

N₍_x₎_AN₍_y₎

AN₍_x_{) +}_AN₍_y₎₋_AN₍_x₎_AN₍_y₎

for allxy∈E

Example 3.1: Consider a D-Bipolar-fuzzy graph overV = a, b, c, d defined by

A={ j −0.6,

k −0.7,

l −0.9,

m −0.4}

B={ jk −0.47,

jm −0.3,

(7)

F igure6 Strong dombi bipolar fuzzy graph

Definition 3.2:A D-Bipolar-fuzzy graphG= (A, B)is said to be a complete D-Bipolar-fuzzy graph of G∗= (V, E)if

BP(xy) = A

AP₍_x_{) +}_AP₍_y₎₋_AP₍_x₎_AP₍_y₎

and

BN(xy) = A

N₍_x₎_AN₍_y₎

AN₍_x_{) +}_AN₍_y₎₋_AN₍_x₎_AN₍_y₎

for allxy∈V

Remark 3.3: Every complete D-Bipolar-fuzzy graph is strong D-Bipolar-fuzzy graph.

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