Bipolar Neutrosophic Graph Structures | Akram | Journal of the Indonesian Mathematical Society 442 946 1 PB

J. Indones. Math. Soc.
Vol. 23, No. 1 (2017), pp. 55–80.

BIPOLAR NEUTROSOPHIC GRAPH STRUCTURES
Muhammad Akram1 and Muzzamal Sitara2
1

Department of Mathematics, University of the Punjab, New Campus,
Lahore, Pakistan,
makrammath@yahoo.com
2
Department of Mathematics, University of the Punjab, New Campus,
Lahore, Pakistan,
muzzamalsitara@gmail.com

Abstract. In this research study, we introduce the concept of bipolar single-valued
neutrosophic graph structures. We discuss certain notions of bipolar single-valued
neutrosophic graph structures with examples. We present some methods of construction of bipolar single-valued neutrosophic graph structures. We also investigate
some of their prosperities.
Key words and Phrases: Graph structure, bipolar single-valued neutrosophic graph
structure, operations.

Abstrak. Pada penelitian ini, kami memperkenalkan konsep struktur graf neutrosofik bipolar bernilai tunggal. Kami mengkaji ide-ide tertentu dari struktur
graf neutrosofik bipolar bernilai tunggal berserta contoh-contohnya. Kami menyajikan beberapa metode konstruksi struktur graf neutrosofik bipolar bernilai tunggal.
Kami juga memeriksa beberapa sifat-sifat mereka.
Kata kunci: Striktur graf, struktur graf neutrosofik bipolar bernilai tunggal, operasi

1. Introduction
Fuzzy graph theory has a number of applications in modeling real time systems where the level of information inherent in the system varies with different
levels of precision. Fuzzy models are becoming useful because of their aim in reducing the differences between the traditional numerical models used in engineering
and sciences and the symbolic models used in expert systems. In 1973, Kauffmann
[13] illustrated the notion of fuzzy graphs based on Zadeh’s fuzzy relations [24].
Rosenfeld [16] discussed several basic graph-theoretic concepts, including bridges,
2000 Mathematics Subject Classification: 03E72, 68R10, 68R05.
Received: 02-02-2017, revised: 02-03-2017, accepted: 03-02-2017.
55

56

M. Akram and M. Sitara

cut-nodes, connectedness, trees and cycles. Later on, Bhattacharya [8] gave some
remarks on fuzzy graphs. In 1994, Mordeson and Chang-Shyh [14] defined some
operations on fuzzy graphs. The complement of fuzzy graph was defined in [14].
Further, this concept was discussed by Sunitha and Vijayakumar [20]. Akram described bipolar fuzzy graphs in 2011 [1]. Akram and Shahzadi [5] described the
concept of neutrosophic soft graphs with applications. Dinesh and Ramakrishnan
[12] introduced the concept of the fuzzy graph structure and investigated some related properties. Akram and Akmal [4] proposed the notion of bipolar fuzzy graph
structures. On the other hand, Dhavaseelan et al. [10] defined strong neutrosophic
graphs. Akram and Sarwar [3] portrayed bipolar neutrosophic graphs with applications. Akram and Shahzadi [5] introduced the notion of neutrosophic soft graphs
with applications. Akram [2] introduced the notion of single-valued neutrosophic
planar graphs. Representation of graphs using intuitionistic neutrosophic soft sets
was discussed in [6]. Single-valued neutrosophic minimum spanning tree and its
clustering method were studied by Ye [22]. In this research study, we introduce the
concept of bipolar single-valued neutrosophic graph structures. We discuss certain
notions of bipolar single-valued neutrosophic graph structures with examples. We
present some methods of construction of bipolar single-valued neutrosophic graph
structures. We also investigate some of their prosperities.

2. BIPOLAR SINGLE-VALUED NEUTROSOPHIC GRAPH
STRUCTURES
Smarandache [19] introduced neutrosophic sets as a generalization of fuzzy sets

and intuitionistic fuzzy sets. A neutrosophic set has three constituents: truthmembership, indeterminacy-membership and falsity-membership, in which each
membership value is a real standard or non-standard subset of the unit interval
]0− , 1+ [. In real-life problems, neutrosophic sets can be applied more appropriately
by using the single-valued neutrosophic sets defined by Smarandache [19] and Wang
et al [21].
Definition 2.1. [19] A neutrosophic set N on a non-empty set V is an object of
the form
N = {(v, TN (v), IN (v), FN (v)) : v ∈ V }
where, TN , IN , FN : V →]0− , 1+ [ and there is no restriction on the sum of TN (v),
IN (v) and FN (v) for all v ∈ V .
Definition 2.2. [21]A single-valued neutrosophic set N on a non-empty set V is
an object of the form
N = {(v, TN (v), IN (v), FN (v)) : v ∈ V }
where, TN , IN , FN : V → [0, 1] and sum of TN (v), IN (v) and FN (v) is confined
between 0 and 3 for all v ∈ V .

Bipolar Neutrosophic Graph Structures

57

Deli et al. [9] defined bipolar neutrosophic sets a generalization of bipolar fuzzy sets.
They also studied some operations and applications in decision making problems.
Definition 2.3. [9]A bipolar single-valued neutrosophic set on a non-empty set V
is an object of the form
N
P
(v), FBN (v)) : v ∈ V }
B = {(v, TBP (v), IB
(v), FBP (v), TBN (v), IB
P
N
where, TBP , IB
, FBP : V → [0, 1] and TBN , IB
, FBN : V → [−1, 0]. The positive values
P
P
P
TB (v), IB (v), FB (v) denote the truth, indeterminacy and falsity membership values
N
of an element v ∈ V , whereas negative values TBN (v), IB

(v), FBN (v) indicates the
implicit counter property of truth, indeterminacy and falsity membership values of
an element v ∈ V .

Definition 2.4. A bipolar single-valued neutrosophic graph on a non-empty set V
is a pair G = (B, R), where B is a bipolar single-valued neutrosophic set on V and
R is a bipolar single-valued neutrosophic relation in V such that
TRP (bd) ≤ TBP (b) ∧ TBP (d),
TRN (bd) ≥ TBN (b) ∨ TBN (d),

P
P
P
IR
(bd) ≤ IB
(b) ∧ IB
(d),
N
N
N

IR
(bd) ≥ IB
(b) ∨ IB
(d),

FRP (bd) ≤ FBP (b) ∨ FBP (d),
FRN (bd) ≥ FBN (b) ∧ FBN (d),

for all b, d ∈ V.
We now define bipolar single-valued neutrosophic graph structure.
ˇ bn = (B, B1 , B2 , . . . , Bm ) is called bipolar single-valued neutroDefinition 2.5. G
ˇ s = (V, V1 , V2 , . . . , Vm ) if
sophic graph structure(BSVNGS) of graph structure G
P
P
P
N
N
N
B =< b, T (b), I (b), F (b), T (b), I (b), F (b) > and

Bk =< (b, d), TkP (b, d), IkP (b, d), FkP (b, d), TkN (b, d), IkN (b, d), FkN (b, d) > are bipolar
single-valued neutrosophic(BSVN) sets on V and Vk , respectively, such that
TkP (b, d) ≤ min{T P (b), T P (d)}, IkP (b, d) ≤ min{I P (b), I P (d)},
FkP (b, d) ≤ max{F P (b), F P (d)}, TkN (b, d) ≥ max{T N (b), T N (d)},
IkN (b, d) ≥ max{I N (b), I N (d)}, FkN (b, d) ≥ min{F N (b), F N (d)},
for all b, d ∈ V. Note that 0 ≤ TkP (b, d) + IkP (b, d) + FkP (b, d) ≤ 3,
−3 ≤ TkN (b, d) + IkN (b, d) + FkN (b, d) ≤ 0 for all (b, d) ∈ Vk .
ˇ s = (V, V1 , V2 ) such that V =
Example 2.6. Consider graph structure(GSR) G
{b1 , b2 , b3 , b4 }, V1 = {b1 b3 , b1 b2 , b3 b4 }, V2 = {b1 b4 , b2 b3 }. By defining bipolar singlevalued neutrosophic sets B, B1 and B2 on V , V1 and V2 , respectively, we can draw
a bipolar SVNGS as depicted in Fig. 1.
ˇ bn = (B, B1 , B2 , . . . , Bm ) be a BSVNGS of GS G
ˇ s . If
Definition 2.7. Let G
′
′
′
′
ˇ bn = (B , B , B , . . . , B ) is a BSVNGS of G
ˇ s such that

H
1
2
m
T ′P (b) ≤ T P (k), I ′P (b) ≤ I P (b), F ′P (b) ≥ F P (b),
T ′N (b) ≥ T P (k), I ′N (b) ≥ I P (b), F ′N (b) ≤ F N (b) ,

M. Akram and M. Sitara

−0.

2, −

0 .3 ,

−0.

4)

0.3)

0.2, − )
0.3, −
.3
, 0.3, − −0.2, −0
.2
0
,
,
bb 4(0.3
)
, −0.3
3
.3
.
0
,
0
.3, 0.2
B1 (0
2, −

−
0.2,

−
0.2,

0.4)

B1 (0.2, 0.2, 0.4, −0.2, −0.2, −0.4)

−
0.4,
0.2,
0.2,
B 2(

b1 (0.2, 0.3, 0.4, −0.2, −0.3, −0.4)
b
B1 (

0 .2 ,
0 .3 ,
0 .4 ,

,
0 .2
2(

0 .2

, 0.

0
3, −

.2 ,

−0

b

.

b3 (0.3, 0.4, 0.3, −0.3, −0.4, −0.3)

58

b
B
b2 (0.2, 0.2, 0.3, −0.2, −0.2, −0.3)

−
0.1,

−0.
1, −

0 .2 ,

−0.

5)

0.4)
0.1, − )
0.2, −
.4
, 0.4, − −0.1, −0
.1
0
,
,
bb 4 (0.2
)
, −0.2
4
.4
.
0
,
0
′
.2, 0.1
B 1(0
1, −

−
0.1,

0.5)

B1′ (0.1, 0.1, 0.5, −0.1, −0.1, −0.5)

−
0.5,
0.1,
′ 0.1,
B 2(

b1 (0.1, 0.2, 0.5, −0.1, −0.2, −0.5)
b
B ′(
1 0.
1, 0
.2 , 0
.5 ,

.
1, 0
, 0.
1
.
′ (0
b
B2
b2 (0.1, 0.1, 0.4, −0.1, −0.1, −0.4)

0
4, −

.1 ,

−0

.

b

b3 (0.2, 0.3, 0.4, −0.2, −0.3, −0.4)

Figure 1. A bipolar single-valued neutrosophic graph structure

Figure 2. A BSVN subgraph structure
Tk′P (b, d) ≤ TkP (b, d), Ik′P (b, d) ≤ IkP (b, d), Fk′P (b, d) ≥ FkP (b, d),
Tk′N (b, d) ≥ TkN (b, d), Ik′N (b, d) ≥ IkN (b, d), Fk′N (b, d) ≤ FkN (b, d),
∀ b ∈ V and (b, d) ∈ Vk , k = 1, 2, . . . , m.
ˇ bn is named as a bipolar single-valued neutrosophic(BSVN) subgraph strucThen H
ˇ bn .
ture of BSVNGS G
ˇ bn = (B ′ , B1′ , B2′ ) of GS G
ˇ s = (V, V1 , V2 )
Example 2.8. Consider a BSVNGS H
ˇ
as depicted in Fig. 2. Routine calculations indicate that Hbn is BSVN subgraphˇ bn .
structure of BSVNGS G
′
ˇ bn = (B ′ , B1′ , B2′ , . . . , Bm
Definition 2.9. A BSVNGS H
) is called a BSVN induced
ˇ
subgraph-structure of BSVNGS Gbn by Q ⊆ V if

59

.2
(0
b1
.3 ,
0 .4
,−
0 .2

0 .4 ,

)
− 0 .4

b

0 .2 ,

0 .2 ,

,0
,−

.2 ,
′
B 1(0

,−
− 0 .2

0 .3

b

b2 (0.2, 0.2, 0.3, −0.2, −0.2, −0.3)

Bipolar Neutrosophic Graph Structures

,−

−
4,
0.

−

−

2,

0.

2,

3,

0.

0.

−

−

−

3,

0.

3,

0.

0.
3

)

B1′ (0.2, 0.3, 0.4, −0.2, −0.3, −0.4)

3)

0.

2,

4,

0.

0.

2,

)
0.4

B′
2(
0.

b3

(0
.

3,

b

Figure 3. A BSVN induced subgraph-structure
T ′P (b) = T P (b), I ′P (b) = I P (b), F ′P (b) = F P (b),
T ′N (b) = T N (b), I ′N (b) = I N (b), F ′N (b) = F N (b),
Tk′P (b, d) = TkP (b, d), Ik′P (b, d) = IkP (b, d), Fk′P (b, d) = FkP (b, d),
Tk′N (b, d) = TkN (b, d), Ik′N (b, d) = IkN (b, d), Fk′N (b, d) = FkN (b, d),
∀ b, d ∈ Q, k = 1, 2, . . . , m.
Example 2.10. A BSVNGS depicted in Fig.3 is a BSVN induced subgraphstructure of BSVNGS represented in Fig. 1.
′
ˇ bn = (B ′ , B1′ , B2′ , . . . , Bm
Definition 2.11. A BSVNGS H
) is called BSVN spanning
ˇ
subgraph-structure of BSVNGS Gbn = (B, B1 , B2 , . . . , Bm ) if B ′ = B and

Tk′P (b, d) ≤ TkP (b, d), Ik′P (b, d) ≤ IkP (b, d), Fk′P (b, d) ≥ FkP (b, d),
≥ TkN (b, d), Ik′N (b, d) ≥ IkN (b, d), Fk′N (b, d) ≤ FkN (b, d), k = 1, 2, . . . , m.

Tk′N (b, d)

Example 2.12. A BSVNGS represented in Fig. 4 is a BSVN spanning subgraphstructure of BSVNGS represented in Fig. 1.
ˇ bn = (B, B1 , B2 , . . . , Bm ) be a BSVNGS. Then bd ∈ Bk is
Definition 2.13. Let G
called a BSVN Bk -edge or shortly Bk -edge, if
TkP (b, d) > 0 or IkP (b, d) > 0 or FkP (b, d) > 0 or TkN (b, d) < 0 or IkN (b, d) < 0 or
FkN (b, d) < 0 or all these conditions are satisfied.
Consequently support of Bk is defined as:

M. Akram and M. Sitara

60

.4, −0

.2, −0

0 .4

0.4)

, 0.2, 0

.1, −0

.3, −0

−0.3, −

−
2,
0.
−
,
4
.1 ,
0.
′ (0
1,
0.B
B1
,
b
.2 1 ′(0
′ (0
.2 4 (0.
,0
B2
3,
.1
, 0 0.2
B′
,0
.3
2 (0
,
.1 ,
− .3,
0.
0 .1
2, −0
, 0.
− .3,
3, −
0.
1, −0
0 .1
− .2
,−
0 ,−
0 .1
, − .3) 0.3
0.3
)
)

,
0 .1

b

.4)
.2, −0

, 0.2, 0

4)
0.

b

b2 (0.2

−
1,
0.

, −0.2,

′

−0

.1 ,

.4 )
−0

B 1(0.1

,−

,
0 .1

0.3, 0.4

b

.2, −0
.3)

b1 (0.2,

b

b

.3 ,
(0
3

0 .4

,0

−
.3 ,

0 .3

,−

,
0 .4

.3 )
−0

Figure 4. A BSVN spanning subgraph-structure
supp(Bk ) = {bd ∈ Bk : TkP (b, d) > 0} ∪ {bd ∈ Bk : IkP (b, d) > 0} ∪ {bd ∈ Bk :
FkP (b, d) > 0} ∪ {bd ∈ Bk : TkN (b, d) < 0} ∪ {bd ∈ Bk : IkN (b, d) < 0} ∪ {bd ∈ Bk :
FkN (b, d) < 0}, k = 1, 2, . . . , m.
ˇ bn = (B, B1 , B2 , . . . , Bm ) is a sequence
Definition 2.14. Bk -path in BSVNGS G
b1 , b2 , . . . , bm of distinct nodes(vertices) (except bm = b1 ) in V such that bk−1 bk is
a BSVN Bk -edge for all k = 2, . . . , m.
ˇ bn = (B, B1 , B2 , . . . , Bm ) is Bk -strong for any
Definition 2.15. A BSVNGS G
k ∈ {1, 2, . . . , m} if
TkP (b, d) = min{T P (b), T P (d)}, IkP (b, d) = min{I P (b), I P (d)},
FkP (b, d) = max{F P (b), F P (d)}, TkN (b, d) = max{T N (b), T N (d)},
IkN (b, d) = max{I N (b), I N (d)}, FkN (b, d) = min{F N (b), F N (d)},
ˇ bn is Bk -strong for all k ∈ {1, 2, . . . , m}, then G
ˇ bn is called
∀ bd ∈ supp(Bk ). If G
strong BSVNGS.
ˇ bn = (B, B1 , B2 , B3 ) as depicted in Fig. 5.
Example 2.16. Consider BSVNGS G
ˇ bn is strong BSVNGS, since it is B1 −, B2 − and B3 − strong.
Then G
ˇ bn = (B, B1 , B2 , . . . , Bm ) is called complete BSVNGS
Definition 2.17. A BSVNGS G
if

Bipolar Neutrosophic Graph Structures

61

,
)
0.2
0.6
, − 0.6, −
3
.
,
−0
0.3
.3, 0.2,
−0 B 1(
,
.6
b
,0
0.3

b

2,
(0.

, 0.
0.1

3,

−
.2,
−0

.6)
−0

b

b

,
0.1

.3)
−0

B2

(0.
B1

b

.3,
−0

.3,
3, 0

,−
0.5

b

.5)
−0 , −
.2, 2, 0.4
0
, − 0.
0.3 .3,
, − b 2(0
0.5

,
0.3

.3,
−0

,
0.2
,−
0.3

B

b5 (0.2, 0.1, 0.3, −0.2, −0.1, −0.3)
B
B2 (0.2, 0.1, 0.5, −0.2, −0.1, −0.5)
1(
0.2
B
,0
3(
.1,
0.3
b3
0.6
(0 , 0.3
,−
.3,
,0
0.2
0.3 .5,
,−
−
,0
0
0.1
.3,
.3,
,−
−0
−0
0.6
.3,
.3,
)
−0
−0
.5)
.3,
−0
−0.6)
,
−0.3
,
−0.3
0.6,
0.3,
,
B2 (0.3
.3)

3,
(0.
b6

B
2
, −0.4) b7 (0.2,
B1 (0.3, 0.2, 0.4, −0.3, −0.2
(0
.4, 0.3,
3(
0.3
0.4 0.4
,
,0
,0
.2,
.3, −0.
−0 2, −
0.5
.4,
,−
0
−0 .3, −
0.3
.4,
,−
0
−0 .4)
0.2
.3)
,−
0.5
−0.5)
−0.3,
−0.2,
B1 (0.2, 0.3, 0.5,
)
b4 (0.4, 0.3, 0.5, −0.4, −0.3, −0.5)

b8 (0.2, 0.3, 0.4, −0.2, −0.3, −0.4)

.5)
−0

b

.4)
−0

.2,
3, 0
(0.
B1
b1 (0.4, 0.3, 0.5, −0.4, −0.3, −0.5)
b

Figure 5. A Strong BSVNGS

(1)
(2)
(3)

ˇ bn is strong BSVNGS.
G
supp(Bk ) 6= ∅, for all k = 1, 2, . . . , m.
For all b, d ∈ V , bd is a Bk − edge for some k.

ˇ bn = (B, B1 , B2 ) be BSVNGS of GS G
ˇ = (V, V1 , V2 ), such
Example 2.18. Let G
that V = {b1 , b2 , b3 , b4 }, V1 = {b1 b2 , b3 b4 }, V2 = {b1 b3 , b2 b3 , b1 b4 , b2 b4 }.
ˇ bn is strong BSVNGS. MoreThrough direct calculations it is easily shown that G
over, supp(B1 ) 6= ∅, supp(B2 ) 6= ∅ and each pair bk bl of nodes in V is either a
ˇ bn is complete BSVNGS, that is, B1 − B2 −complete
B1 -edge or B2 -edge. Hence G
BSVNGS.
ˇ b1 = (B1 , B11 , B12 , . . . , B1m ) and G
ˇ b2 = (B2 , B21 , B22 , . . . , B2m )
Definition 2.19. Let G
ˇ
ˇ
be two BSVNGSs. Lexicographic product of Gb1 and Gb2 , denoted by
ˇ b1 • G
ˇ b2 = (B1 • B2 , B11 • B21 , B12 • B22 , . . . , B1m • B2m ),
G
is defined as:

M. Akram and M. Sitara

62

0.5
)

b1 (0.3, 0.4, 0.5, −0.3, −0.4, −0.5)
b

0 .2
,−
0.2

5, −

, 0.

0 .2

0.5
)

0.4
,−
0.1
,−
(0.
1, 0
.3,
0.5
,−

,−
0 .2
,−
)
0.5

b

B2

B

2 (0

.2 ,

,−
0.3

B2 (0.2, 0.2, 0.4, −0.2, −0.2, −0.4) b

B2 (0.1, 0.4, 0.5, −0.1, −0.4, −0.5)

,−
0.2

b2 (0.2, 0.3, 0.3, −0.2, −0.3, −0.3)
)
0.5

,−
0.5
.3,
2, 0
(0.

,−

B1

0 .1
,−
0 .5
.2 ,
1, 0
(0.
B1
b4 (0.2, 0.2, 0.4, −0.2, −0.2, −0.4)
b

b3 (0.1, 0.4, 0.5, −0.1, −0.4, −0.5)

Figure 6. A complete BSVNGS

(i)

(ii)

(iii)

(iv)

(v)


P
P
P
P
P

 T(B1 •B2 ) (bd) = (TB1 • TB2 )(bd) = TB1 (b) ∧ TB2 (d)
P
P
P
P
P
I(B1 •B2 ) (bd) = (IB1 • IB2 )(bd) = IB1 (b) ∧ IB2 (d)

 FP
(bd) = (FBP1 • FBP2 )(bd) = FBP1 (b) ∨ FBP2 (d)
 (B1 •B2 )
N
N
N
N
N

 T(B1 •B2 ) (bd) = (TB1 • TB2 )(bd) = TB1 (b) ∨ TB2 (d)
N
N
N
N
N
I(B
(bd) = (IB
• IB
)(bd) = IB
(b) ∨ IB
(d)
1
2
1
2
1 •B2 )

N
N
N
N
 FN
(bd)
=
(F
•
F
)(bd)
=
F
(b)
∧
F
B1
B2
B1
B2 (d)
(B1 •B2 )
for all (bd) ∈ V1 × V2 ,

P
 T(B
(bd1 )(bd2 ) = (TBP1k • TBP2k )(bd1 )(bd2 ) = TBP1 (b) ∧ TBP2k (d1 d2 )

1k •B2k )
P
P
P
P
P
I(B1k •B2k ) (bd1 )(bd2 ) = (IB
• IB
)(bd1 )(bd2 ) = IB
(b) ∧ IB
(d1 d2 )
1
1k
2k
2k

P
P
P
P
 FP
(bd
)(bd
)
=
(F
•
F
)(bd
)(bd
)
=
F
(b)
∨
F
1
2
1
2
B1k
B2k
B1
B2k (d1 d2 )
 (B1k •B2k )
N
N
N
N
N

 T(B1k •B2k ) (bd1 )(bd2 ) = (TB1k • TB2k )(bd1 )(bd2 ) = TB1 (b) ∨ TB2k (d1 d2 )
N
N
N
N
N
I(B1k •B2k ) (bd1 )(bd2 ) = (IB1k • IB2k )(bd1 )(bd2 ) = IB1 (b) ∨ IB2k (d1 d2 )

N
N
N
N
 FN
(B1k •B2k ) (bd1 )(bd2 ) = (FB1k • FB2k )(bd1 )(bd2 ) = FB1 (b) ∧ FB2k (d1 d2 )
for
 all b ∈ V1 , (d1 d2 ) ∈ V2k ,
P
P
P
P
P

 T(B1k •B2k ) (b1 d1 )(b2 d2 ) = (TB1k • TB2k )(b1 d1 )(b2 d2 ) = TB1k (b1 b2 ) ∧ TB2k (d1 d2 )
P
P
P
P
P
I(B1k •B2k ) (b1 d1 )(b2 d2 ) = (IB1k • IB2k )(b1 d1 )(b2 d2 ) = IB1k (b1 b2 ) ∧ IB2k (d1 d2 )

P
P
P
P
 FP
(B1k •B2k ) (b1 d1 )(b2 d2 ) = (FB1k • FB2k )(b1 d1 )(b2 d2 ) = FB1k (b1 b2 ) ∨ FB2k (d1 d2 )

Bipolar Neutrosophic Graph Structures

B1

−0
.5)
−0
.2,
0.5
,

−0
.3,

0.6)

0.2
,
.3,

B2

0.4)

2 (0

−0
.8)
−0
.1,
−0
.5,
0.8
,
.5,

−
0.1,

−
0.2,

1 (0

−
0.2,

−
0.4,

0.1
,

,−
, 0.4
, 0.1
(0.2

,−
, 0.6
, 0.2
(0.4

b

b

d3 (0.5, 0.3, 0.5, −0.5, −0.3, −0.5)

b

B 21

d1 (0.2, 0.1, 0.3, −0.2, −0.1, −0.3)

b

B 12

b3 (0.4, 0.2, 0.4, −0.4, −0.2, −0.4)
b1 (0.5, 0.1, 0.6, −0.5, −0.1, −0.6)
b

63

b

b

b2 (0.5, 0.2, 0.8, −0.5, −0.2, −0.8) b4 (0.5, 0.2, 0.6, −0.5, −0.2, −0.6)

d2 (0.3, 0.2, 0.4, −0.3, −0.2, −0.4)

ˇ b1 and G
ˇ b2
Figure 7. Two BSVNGSs G
b2

,−
0.6

B11 • B21 (0.2, 0.1, 0.8, −0.2, −0.1, −0.8)

b

0.2,
8, −
1, 0.
.
0
,
.2
d 1(0

.8)
, −0
−0.1

b

b2

B11 • B21 (0.2, 0.1, 0.6, −0.2, −0.1, −0.6)
B1

2

b1

d1

(0
.2,

•B
2

B1
22 (
0.3
,

•B

−
8,
0.
2,
.
,0
.3
(0
2

2

d3

(0
.5,

0.2
,0
8) .8, −
.
0
0.5
−
2,
,−
0. b
0.2
−
,−
3,
.
0
0.
B11 • B21 (0.2, 0.1, 0.8, −0.2, −0.1, −0.8)

d
b1

,
0.1
3,
(0.
2

,
0.3

.1,
−0

.6)
−0

8)

0.1
, 0.
6, −
0.3
)
.
3
, 0.2
0.1
,−
0.6
, 0.8
0.1
b
b
,0
,−
,− b
, −0
1
.
.6,
0
0.6
.3, −
)
−0 B11 • B21 (0.2, 0.1, 0.8, −0.2, −0.1, −0.8)
,−
0.2,
0.5
.2,
−0.
,−
8)
−0
6
.
0
.1,
.1,
−0
,0
0.5
.6)
(
3
b 1d
b2 d

2 (0


N
N
N
N
N

 T(B1k •B2k ) (b1 d1 )(b2 d2 ) = (TB1k • TB2k )(b1 d1 )(b2 d2 ) = TB1k (b1 b2 ) ∨ TB2k (d1 d2 )
N
N
N
N
N
I(B
(b1 d1 )(b2 d2 ) = (IB
• IB
)(b1 d1 )(b2 d2 ) = IB
(b1 b2 ) ∨ IB
(d1 d2 )
(vi)
1k
2k
1k
2k
1k •B2k )

N
N
N
N
 FN
(b
d
)(b
d
)
=
(F
•
F
)(b
d
)(b
d
)
=
F
(b
b
)
∧
F
1
1
2
2
1
1
2
2
1
2
B1k
B2k
B1k
B2k (d1 d2 )
(B1k •B2k )
for all (b1 b2 ) ∈ V1k , (d1 d2 ) ∈ V2k .
ˇ b1 = (B1 , B11 , B12 ) and G
ˇ b2 = (B2 , B21 , B22 ) are two
Example 2.20. Consider G
ˇ s1 = (V1 , V11 , V12 ) and G
ˇ s2 = (V2 , V21 , V22 ), respectively, as
BSVNGSs of GSs G
depicted in Fig. 7, where V11 = {b1 b2 }, V12 = {b3 b4 }, V21 = {d1 d2 }, V22 = {d2 d3 }.
ˇ b1 and G
ˇ b2 shown in Fig. 7 is defined as:
Lexicographic product of BSVNGSs G
ˇ b1 • G
ˇ b2 = {B1 • B2 , B11 • B21 , B12 • B22 } and is depicted in Fig. 8.
G

M. Akram and M. Sitara

64

b4

.6)
−0

d3

(0
.5,

0.2
,0
3, 0.
.6,
2, 0.
−0
B12 • B22 (0.3, 0.2, 0.6, −0.3, −0.2, −0.6)
4, −
.5,
0
b
b
b
.
3
−
, −0
−0
,
6
.
2, −
.
.2,
0.4)
,0
4)
1
.
−0
.
0
0
,
.6)
−
2
,
.
1
0
.
(
0
d1
,−
b4
B
•
B
(0.3,
0.2,
12
0.6,
−0.3,
22
2
−0.2, −0.6)
0.
B1
,−
1 •
.4
B2
0
1 (0
1,
.2,
0.
0.1
2,
, 0.
0.
6, −
(
0.2
21
,−
B
0.1
•
)
,−
.6)
1
.5
0
1
0.6
−
b3 d
,
−0
B
2
.
)
0
b
b
,−
.2,
b
1 (0
4
.
0
0
−
.2,
−
B12 • B22 (0.3, 0.2, 0.6, −0.3, −0.2, −0.6)
0.5,
0.1
.3,
0.2,
, 0.
−0
(0.4,
,
4, −
6
b
3d3
.
0.2
,0
,−
0.2
0.1
.3,
,−
0
(
0.4
d2
)
b4
2 (0.

B12 • B22 (0.3, 0.2, 0.5, −0.3, −0.2, −0.5)

,−
0.2

,
0.1

b3 d

ˇ b1 • G
ˇ b2
Figure 8. G
ˇ b1 • G
ˇ b2 = (B1 • B2 , B11 • B21 , B12 •
Theorem 2.21. Lexicographic product G
B22 , . . . , B1m • B2m ) of two BSVNSGSs of GSs Gˇs1 and Gˇs2 is a BSVNGS of
Gˇs1 • Gˇs2 .
proof. Consider two cases:
Case 1.: For b ∈ V1 , d1 d2 ∈ V2k
P
((bd1 )(bd2 )) = TBP1 (b) ∧ TBP2k (d1 d2 )
T(B
1k •B2k )

≤ TBP1 (b) ∧ [TBP2 (d1 ) ∧ TBP2 (d2 )]
= [TBP1 (b) ∧ TBP2 (d1 )] ∧ [TBP1 (b) ∧ TBP2 (d2 )]
P
P
(bd2 ),
(bd1 ) ∧ T(B
= T(B
1 •B2 )
1 •B2 )
N
T(B
((bd1 )(bd2 )) = TBN1 (b) ∨ TBN2k (d1 d2 )
1k •B2k )

≥ TBN1 (b) ∨ [TBN2 (d1 ) ∨ TBN2 (d2 )]
= [TBN1 (b) ∨ TBN2 (d1 )] ∨ [TBN1 (b) ∨ TBN2 (d2 )]
N
N
= T(B
(bd1 ) ∨ T(B
(bd2 ),
1 •B2 )
1 •B2 )

P
P
P
(b) ∧ IB
(d1 d2 )
((bd1 )(bd2 )) = IB
I(B
1
2k
1k •B2k )
P
P
P
(d2 )]
(d1 ) ∧ IB
(b) ∧ [IB
≤ IB
2
2
1
P
P
P
P
(d2 )]
(b) ∧ IB
(d1 )] ∧ [IB
(b) ∧ IB
= [IB
2
1
2
1
P
P
= I(B
(bd1 ) ∧ I(B
(bd2 ),
1 •B2 )
1 •B2 )

Bipolar Neutrosophic Graph Structures
N
N
N
(b) ∨ IB
(d1 d2 )
I(B
((bd1 )(bd2 )) = IB
1
2k
1k •B2k )
N
N
N
(d2 )]
(d1 ) ∨ IB
(b) ∨ [IB
≥ IB
2
2
1
N
N
N
N
(d2 )]
(b) ∨ IB
(d1 )] ∨ [IB
(b) ∨ IB
= [IB
2
1
2
1
N
N
= I(B
(bd1 ) ∨ I(B
(bd2 ),
1 •B2 )
1 •B2 )

P
((bd1 )(bd2 )) = FBP1 (b) ∨ FBP2k (d1 d2 )
F(B
1k •B2k )

≤ FBP1 (b) ∨ [FBP2 (d1 ) ∨ FBP2 (d2 )]
= [FBP1 (b) ∨ FBP2 (d1 )] ∨ [FBP1 (b) ∨ FBP2 (d2 )]
P
P
= F(B
(bd1 ) ∨ F(B
(bd2 ),
1 •B2 )
1 •B2 )

N
F(B
((bd1 )(bd2 )) = FBN1 (b) ∧ FBN2k (d1 d2 )
1k •B2k )

≥ FBN1 (b) ∧ [FBN2 (d1 ) ∧ FBN2 (d2 )]
= [FBN1 (b) ∧ FBN2 (d1 )] ∧ [FBN1 (b) ∧ FBN2 (d2 )]
N
N
(bd2 ),
(bd1 ) ∧ F(B
= F(B
1 •B2 )
1 •B2 )

for bd1 , bd2 ∈ V1 • V2 .
Case 2.: For b1 b2 ∈ V1k , d1 d2 ∈ V2k
P
((b1 d1 )(b2 d2 )) = TBP1k (b1 b2 ) ∧ TBP2k (d1 d2 )
T(B
1k •B2k )

≤ [TBP1 (b1 ) ∧ TBP1 (b2 ] ∧ [TBP2 (d1 ) ∧ TBP2 (d2 )]
= [TBP1 (b1 ) ∧ TBP2 (d1 )] ∧ [TBP1 (b2 ) ∧ TBP2 (d2 )]
P
P
= T(B
(b1 d1 ) ∧ T(B
(b2 d2 ),
1 •B2 )
1 •B2 )

N
T(B
((b1 d1 )(b2 d2 )) = TBN1k (b1 b2 ) ∨ TBN2k (d1 d2 )
1k •B2k )

≥ [TBN1 (b1 ) ∨ TBN1 (b2 ] ∨ [TBN2 (d1 ) ∨ TBN2 (d2 )]
= [TBN1 (b1 ) ∨ TBN2 (d1 )] ∨ [TBN1 (b2 ) ∨ TBN2 (d2 )]
N
N
(b2 d2 ),
(b1 d1 ) ∨ T(B
= T(B
1 •B2 )
1 •B2 )

P
P
P
((b1 d1 )(b2 d2 )) = IB
(b1 b2 ) ∧ IB
(d1 d2 )
I(B
1k
2k
1k •B2k )
P
P
P
P
(d2 )]
(d1 ) ∧ IB
(b2 ] ∧ [IB
(b1 ) ∧ IB
≤ [IB
2
2
1
1
P
P
P
P
(d2 )]
(b2 ) ∧ IB
(d1 )] ∧ [IB
(b1 ) ∧ IB
= [IB
2
1
2
1
P
P
= I(B
(b1 d1 ) ∧ I(B
(b2 d2 ),
1 •B2 )
1 •B2 )

65

66

M. Akram and M. Sitara
N
N
N
(d1 d2 )
I(B
((b1 d1 )(b2 d2 )) = IB
(b1 b2 ) ∨ IB
2k
1k
1k •B2k )
N
N
N
N
(d2 )]
(d1 ) ∨ IB
(b2 ] ∨ [IB
(b1 ) ∨ IB
≥ [IB
2
2
1
1
N
N
N
N
(d2 )]
(b2 ) ∨ IB
(d1 )] ∨ [IB
(b1 ) ∨ IB
= [IB
2
1
2
1
N
N
(b2 d2 ),
= I(B
(b1 d1 ) ∨ I(B
1 •B2 )
1 •B2 )
P
((b1 d1 )(b2 d2 )) = FBP1k (b1 b2 ) ∨ FBP2k (d1 d2 )
F(B
1k •B2k )

≤ [FBP1 (b1 ) ∨ FBP1 (b2 ] ∨ [FBP2 (d1 ) ∨ FBP2 (d2 )]
= [FBP1 (b1 ) ∨ FBP2 (d1 )] ∨ [FBP1 (b2 ) ∨ FBP2 (d2 )]
P
P
= F(B
(b1 d1 ) ∨ F(B
(b2 d2 ),
1 •B2 )
1 •B2 )
N
((b1 d1 )(b2 d2 )) = FBN1k (b1 b2 ) ∧ FBN2k (d1 d2 )
F(B
1k •B2k )

≥ [FBN1 (b1 ) ∧ FBN1 (b2 ] ∧ [FBN2 (d1 ) ∧ FBN2 (d2 )]
= [FBN1 (b1 ) ∧ FBN2 (d1 )] ∧ [FBN1 (b2 ) ∧ FBN2 (d2 )]
N
N
(b2 d2 ),
(b1 d1 ) ∧ F(B
= F(B
1 •B2 )
1 •B2 )

b1 d1 , b2 d2 ∈ V1 • V2 and h ∈ {1, 2, . . . , m}. This completes the proof.
ˇ b1 = (B1 , B11 , B12 , . . . , B1m ) and G
ˇ b2 = (B2 , B21 , B22 , . . . , B2m )
Definition 2.22. Let G
ˇ b1 and G
ˇ b2 , denoted by
be two BSVNGSs. Strong product of G
ˇ b1 ⊠ G
ˇ b2 = (B1 ⊠ B2 , B11 ⊠ B21 , B12 ⊠ B22 , . . . , B1m ⊠ B2m ),
G
is defined
 as:
P
 T(B
(bd) = (TBP1 ⊠ TBP2 )(bd) = TBP1 (b) ∧ TBP2 (d)

1 ⊠B2 )
P
P
P
P
P
I(B1 ⊠B2 ) (bd) = (IB
⊠ IB
)(bd) = IB
(b) ∧ IB
(d)
(i)
1
2
1
2

P
P
P
P
P
 F
(bd)
=
(F
⊠
F
)(bd)
=
F
(b)
∨
F
B1
B2
B1
B2 (d)
 (B1 ⊠B2 )
N
N
N
N
N

 T(B1 ⊠B2 ) (bd) = (TB1 ⊠ TB2 )(bd) = TB1 (b) ∨ TB2 (d)
N
N
N
N
N
I(B
(bd) = (IB
⊠ IB
)(bd) = IB
(b) ∨ IB
(d)
(ii)
1
2
1
2
1 ⊠B2 )

N
N
N
P
 FN
(bd)
=
(F
⊠
F
)(bd)
=
F
(b)
∧
F
B1
B2
B1
B2 (d)
(B1 ⊠B2 )
for all (bd) ∈ V1 × V2 ,

P
 T(B
(bd1 )(bd2 ) = (TBP1k ⊠ TBP2k )(bd1 )(bd2 ) = TBP1 (b) ∧ TBP2k (d1 d2 )

1k ⊠B2k )
P
P
P
P
P
I(B1k ⊠B2k ) (bd1 )(bd2 ) = (IB
⊠ IB
)(bd1 )(bd2 ) = IB
(b) ∧ IB
(d1 d2 )
(iii)
1
1k
2k
2k

P
P
P
P
 FP
(bd
)(bd
)
=
(F
⊠
F
)(bd
)(bd
)
=
F
(b)
∨
F
1
2
1
2
B1k
B2k
B1
B2k (d1 d2 )
 (B1k ⊠B2k )
N
N
N
N
N

 T(B1k ⊠B2k ) (bd1 )(bd2 ) = (TB1k ⊠ TB2k )(bd1 )(bd2 ) = TB1 (b) ∨ TB2k (d1 d2 )
N
N
N
N
N
I(B1k ⊠B2k ) (bd1 )(bd2 ) = (IB1k ⊠ IB2k )(bd1 )(bd2 ) = IB1 (b) ∨ IB2k (d1 d2 )
(iv)

N
N
N
N
 FN
(B1k ⊠B2k ) (bd1 )(bd2 ) = (FB1k ⊠ FB2k )(bd1 )(bd2 ) = FB1 (b) ∧ FB2k (d1 d2 )
for all b ∈ V1 , (d1 d2 ) ∈ V2k ,

P
P
P
P
P

 T(B1k ⊠B2k ) (b1 d)(b2 d) = (TB1k ⊠ TB2k )(b1 d)(b2 d) = TB2 (d) ∧ TB1k (b1 b2 )
P
P
P
P
P
I(B1k ⊠B2k ) (b1 d)(b2 d) = (IB1k ⊠ IB2k )(b1 d)(b2 d) = IB2 (d) ∧ IB2k (b1 b2 )
(v)

P
P
P
P
 FP
(B1k ⊠B2k ) (b1 d)(b2 d) = (FB1k ⊠ FB2k )(b1 d)(b2 d) = FB2 (d) ∨ FB2k (b1 b2 )

Bipolar Neutrosophic Graph Structures

d1

(0
.2
,0
.1
,0
.8
,−

B11 ⊠ B21 (0.2, 0.1, 0.8, −0.2, −0.1, −0.8)

b2

0.
2,
−

6)
0.

B11 ⊠ B21 (0.2, 0.1, 0.6, −0.2, −0.1, −0.6)

B1

1

B1

1

b

b2 d3 (0.5, 0.2, 0.8, −0.5, −0.2, −0.8)
21 (
b
0

⊠B

⊠B

21 (
0

.2,

b

B 11

.5,

0.1
,

⊠B

0.8
,

,
0.2
1(

2

−0
.2,

, 0.
0.1

0.1
,

0.8
,

−0
.1,

8,

−0
.5,

−0
.1,

−0
.8)

.2,
−0

.1,
−0

−

−0
.8)

)
0.8

,−
.6
,0
1
.
,0
.3
(0

−
3,
0.

−
1,
0.

6)
0.

b

2

B

12

⊠

B2

b1 d3 (0.5, 0.1, 0.6, −0.5, −0.1, −0.6)

0.
1,
−

b

b

6)
0.
−
1,
0.
−
3,
0.
,−
.6
, 0 B12 ⊠ B22 (0.5, 0.1, 0.8, −0.5, −0.1, −0.8)
.1
,0
.3
(0
d2
B11 ⊠ B21 (0.3, 0.1, 0.8, −0.3, −0.1, −0.8)
b1

d
b1

,−
.6
,0
.1
0
,
.2
(0
1

−
2,
0.

−
1,
0.

67

B11 ⊠ B21 (0.2, 0.1, 0.8, −0.2, −0.1, −0.8)

0.
8)

d
b2

−
8,
0.
2,
.
,0
.3
(0
2

−
3,
0.

−
2,
0.

8)
0.

ˇ b1 ⊠ G
ˇ b2
Figure 9. G

N
N
N
N
N

 T(B1k ⊠B2k ) (b1 d)(b2 d) = (TB1k ⊠ TB2k )(b1 d)(b2 d) = TB2 (d) ∨ TB1k (b1 b2 )
N
N
N
N
N
I(B1k ⊠B2k ) (b1 d)(b2 d) = (IB1k ⊠ IB2k )(b1 d)(b2 d) = IB2 (d) ∨ IB2k (b1 b2 )
(vi)

N
N
N
N
 FN
(B1k ⊠B2k ) (b1 d)(b2 d) = (FB1k ⊠ FB2k )(b1 d)(b2 d) = FB2 (d) ∧ FB2k (b1 b2 )
for
 all d ∈ V2 , (b1 b2 ) ∈ V1k .
P
P
P
P
P

 T(B1k ⊠B2k ) (b1 d1 )(b2 d2 ) = (TB1k ⊠ TB2k )(b1 d1 )(b2 d2 ) = TB1k (b1 b2 ) ∧ TB2k (d1 d2 )
P
P
P
P
P
I(B
(b1 d1 )(b2 d2 ) = (IB
⊠ IB
)(b1 d1 )(b2 d2 ) = IB
(b1 b2 ) ∧ IB
(d1 d2 )
(vii)
1k
2k
1k
2k
1k ⊠B2k )

P
P
P
P
 FP
(b
d
)(b
d
)
=
(F
⊠
F
)(b
d
)(b
d
)
=
F
(b
b
)
∨
F
1
1
2
2
1
1
2
2
1
2
B1k
B2k
B1k
B2k (d1 d2 )
 (B1k ⊠B2k )
N
N
N
N
N

 T(B1k ⊠B2k ) (b1 d1 )(b2 d2 ) = (TB1k ⊠ TB2k )(b1 d1 )(b2 d2 ) = TB1k (b1 b2 ) ∨ TB2k (d1 d2 )
N
N
N
N
N
I(B
(b1 d1 )(b2 d2 ) = (IB
⊠ IB
)(b1 d1 )(b2 d2 ) = IB
(b1 b2 ) ∨ IB
(d1 d2 )
(viii)
1k
2k
1k
2k
1k ⊠B2k )

N
N
N
N
 FN
(B1k ⊠B2k ) (b1 d1 )(b2 d2 ) = (FB1k ⊠ FB2k )(b1 d1 )(b2 d2 ) = FB1k (b1 b2 ) ∧ FB2k (d1 d2 )
for all (b1 b2 ) ∈ V1k , (d1 d2 ) ∈ V2k .
ˇ b1 and G
ˇ b2 shown in Fig. 7 is
Example 2.23. Strong product of BSVNGSs G
ˇ b1 ⊠ G
ˇ b2 = {B1 ⊠ B2 , B11 ⊠ B21 , B12 ⊠ B22 } and is depicted in Fig. 9
defined as G
and 10.
ˇ b1 ⊠G
ˇ b2 = (B1 ⊠B2 , B11 ⊠B21 , B12 ⊠B22 , . . . , B1m ⊠
Theorem 2.24. Strong product G
B2m ) of two BSVNGSs of GSs Gˇs1 and Gˇs2 is a BSVNGS of Gˇs1 ⊠ Gˇs2 .
proof.Consider three cases:

M. Akram and M. Sitara
b4

B12 ⊠ B22 (0.3, 0.2, 0.6, −0.3, −0.2, −0.6)

2

⊠B

B 12

B1

2

B1

2

,
0.2

,
0.3

.6)
−0

2

⊠B

⊠B

22 (
0

.3, 0
.2,

0.6
,−
2, 0
0.3
.1, 0
, −0
.6, −
.2, −
0.2,
0.6
−0.
)
1
,
−
−0.6)
−0.1,
−0.2,
0.6,
0.1,
0
(0.2,
d
b4 1
.6)
22 (0
.

B12 ⊠ B22 (0.3, 0.2, 0.5, −0.3, −0.2, −0.5)

0.
4)

b3

d3

B11 ⊠ B21 (0.2, 0.1, 0.6, −0.2, −0.1, −0.6)

0.
2,
−

b

0.
3,
−

1

B1

−
2,
0.

,
0.3
2(

,−
0.6

b

(0
.3
,0
.2
,0
.4
,−

⊠

B

,−
.4
,0
.1
0
,
.2
(0
1

−
1,
0.

4)
0.

.2,
−0

(0
.5
,0
.2
,0
.6
,−

b

d2

b3 d1 (0.2, 0.1, 0.4, −0.2, −0.1, −0.4)
b

b3

B12 × B22 (0.3, 0.2, 0.6, −0.3, −0.2, −0.6)

b

b4

,
0.3

d3

b

d

,
0.2
.3,
0
(
2

,−
0.6

.2,
−0

.6)
−0

ˇ b1 ⊠ G
ˇ b2
Figure 10. G

Case 1.: For b ∈ V1 , d1 d2 ∈ V2k
P
((bd1 )(bd2 )) = TBP1 (b) ∧ TBP2k (d1 d2 )
T(B
1k ⊠B2k )

≤ TBP1 (b) ∧ [TBP2 (d1 ) ∧ TBP2 (d2 )]
= [TBP1 (b) ∧ TBP2 (d1 )] ∧ [TBP1 (b) ∧ TBP2 (d2 )]
P
P
(bd2 ),
(bd1 ) ∧ T(B
= T(B
1 ⊠B2 )
1 ⊠B2 )

N
T(B
((bd1 )(bd2 )) = TBN1 (b) ∨ TBN2k (d1 d2 )
1k ⊠B2k )

≥ TBN1 (b) ∨ [TBN2 (d1 ) ∨ TBN2 (d2 )]
= [TBN1 (b) ∨ TBN2 (d1 )] ∨ [TBN1 (b) ∨ TBN2 (d2 )]
N
N
(bd2 ),
(bd1 ) ∨ T(B
= T(B
1 ⊠B2 )
1 ⊠B2 )

P
P
P
(b) ∧ IB
(d1 d2 )
((bd1 )(bd2 )) = IB
I(B
1
2k
1k ⊠B2k )
P
P
P
(d2 )]
(d1 ) ∧ IB
(b) ∧ [IB
≤ IB
2
2
1
P
P
P
P
(d2 )]
(b) ∧ IB
(d1 )] ∧ [IB
(b) ∧ IB
= [IB
2
1
2
1
P
P
= I(B
(bd1 ) ∧ I(B
(bd2 ),
1 ⊠B2 )
1 ⊠B2 )

0.
5,
−

(0
.4
,0
.2
B12 ⊠ B22 (0.4, 0.2, 0.6, −0.4, −0.2, −0.6)
,0
.5
,−
0.
4,
−
0.
2,
−
0.
5)

68

0.
2,
−

0.
6)

Bipolar Neutrosophic Graph Structures
N
N
N
(b) ∨ IB
(d1 d2 )
I(B
((bd1 )(bd2 )) = IB
1
2k
1k ⊠B2k )
N
N
N
(d2 )]
(d1 ) ∨ IB
(b) ∨ [IB
≥ IB
2
2
1
N
N
N
N
(d2 )]
(b) ∨ IB
(d1 )] ∨ [IB
(b) ∨ IB
= [IB
2
1
2
1
N
N
= I(B
(bd1 ) ∨ I(B
(bd2 ),
1 ⊠B2 )
1 ⊠B2 )

P
((bd1 )(bd2 )) = FBP1 (b) ∨ FBP2k (d1 d2 )
F(B
1k ⊠B2k )

≤ FBP1 (b) ∨ [FBP2 (d1 ) ∨ FBP2 (d2 )]
= [FBP1 (b) ∨ FBP2 (d1 )] ∨ [FBP1 (b) ∨ FBP2 (d2 )]
P
P
= F(B
(bd1 ) ∨ F(B
(bd2 ),
1 ⊠B2 )
1 ⊠B2 )

N
F(B
((bd1 )(bd2 )) = FBN1 (b) ∧ FBN2k (d1 d2 )
1k ⊠B2k )

≥ FBN1 (b) ∧ [FBN2 (d1 ) ∧ FBN2 (d2 )]
= [FBN1 (b) ∧ FBN2 (d1 )] ∧ [FBN1 (b) ∧ FBN2 (d2 )]
N
N
(bd2 ),
(bd1 ) ∧ F(B
= F(B
1 ⊠B2 )
1 ⊠B2 )

for bd1 , bd2 ∈ V1 ⊠ V2 .
Case 2.: For b ∈ V2 , d1 d2 ∈ V1k
P
((d1 b)(d2 b)) = TBP2 (b) ∧ TBP1k (d1 d2 )
T(B
1k ⊠B2k )

≤ TBP2 (b) ∧ [TBP1 (d1 ) ∧ TBP1 (d2 )]
= [TBP2 (b) ∧ TBP1 (d1 )] ∧ [TBP2 (b) ∧ TBP1 (d2 )]
P
P
= T(B
(d1 b) ∧ T(B
(d2 b),
1 ⊠B2 )
1 ⊠B2 )

N
T(B
((d1 b)(d2 b)) = TBN2 (b) ∨ TBN1k (d1 d2 )
1k ⊠B2k )

≥ TBN2 (b) ∨ [TBN1 (d1 ) ∨ TBP1 (d2 )]
= [TBN2 (b) ∨ TBN1 (d1 )] ∨ [TBN2 (b) ∨ TBN1 (d2 )]
N
N
(d2 b),
(d1 b) ∨ T(B
= T(B
1 ⊠B2 )
1 ⊠B2 )

P
P
P
(b) ∧ IB
(d1 d2 )
((d1 b)(d2 b)) = IB
I(B
2
1k
1k ⊠B2k )
P
P
P
(d2 )]
(d1 ) ∧ IB
(b) ∧ [IB
≤ IB
1
1
2
P
P
P
P
(d2 )]
(b) ∧ IB
(d1 )] ∧ [IB
(b) ∧ IB
= [IB
1
2
1
2
P
P
= I(B
(d1 b) ∧ I(B
(d2 b),
1 ⊠B2 )
1 ⊠B2 )

69

70

M. Akram and M. Sitara
N
N
N
(b) ∨ IB
(d1 d2 )
I(B
((d1 b)(d2 b)) = IB
2
1k
1k ⊠B2k )
N
N
N
(d2 )]
(d1 ) ∨ IB
(b) ∨ [IB
≥ IB
1
1
2
N
N
N
N
(d2 )]
(b) ∨ IB
(d1 )] ∨ [IB
(b) ∨ IB
= [IB
1
2
1
2
N
N
= I(B
(d1 b) ∨ I(B
(d2 b),
1 ⊠B2 )
1 ⊠B2 )

P
((d1 b)(d2 b)) = FBP2 (b) ∨ FBP1k (d1 d2 )
F(B
1k ⊠B2k )

≤ FBP2 (b) ∨ [FBP1 (d1 ) ∨ FBP1 (d2 )]
= [FBP2 (b) ∨ FBP1 (d1 )] ∨ [FBP2 (b) ∨ FBP1 (d2 )]
P
P
= F(B
(d1 b) ∨ F(B
(d2 b),
1 ⊠B2 )
1 ⊠B2 )

N
F(B
((d1 b)(d2 b)) = FBN2 (b) ∧ FBN1k (d1 d2 )
1k ⊠B2k )

≥ FBN2 (b) ∧ [FBN1 (d1 ) ∧ FBN1 (d2 )]
= [FBN2 (b) ∧ FBN1 (d1 )] ∧ [FBN2 (b) ∧ FBN1 (d2 )]
N
N
(d2 b),
(d1 b) ∧ F(B
= F(B
1 ⊠B2 )
1 ⊠B2 )

for d1 b, d2 b ∈ V1 ⊠ V2 .
Case 3.: For b1 b2 ∈ V1k , d1 d2 ∈ V2k
P
((b1 d1 )(b2 d2 )) = TBP1k (b1 b2 ) ∧ TBP2k (d1 d2 )
T(B
1k ⊠B2k )

≤ [TBP1 (b1 ) ∧ TBP1 (b2 ] ∧ [TBP2 (d1 ) ∧ TBP2 (d2 )]
= [TBP1 (b1 ) ∧ TBP2 (d1 )] ∧ [TBP1 (b2 ) ∧ TBP2 (d2 )]
P
P
= T(B
(b1 d1 ) ∧ T(B
(b2 d2 ),
1 ⊠B2 )
1 ⊠B2 )

N
T(B
((b1 d1 )(b2 d2 )) = TBN1k (b1 b2 ) ∨ TBN2k (d1 d2 )
1k ⊠B2k )

≥ [TBN1 (b1 ) ∨ TBN1 (b2 ] ∨ [TBN2 (d1 ) ∨ TBN2 (d2 )]
= [TBN1 (b1 ) ∨ TBN2 (d1 )] ∨ [TBN1 (b2 ) ∨ TBN2 (d2 )]
N
N
(b2 d2 ),
(b1 d1 ) ∨ T(B
= T(B
1 ⊠B2 )
1 ⊠B2 )

P
P
P
((b1 d1 )(b2 d2 )) = IB
(b1 b2 ) ∧ IB
(d1 d2 )
I(B
1k
2k
1k ⊠B2k )
P
P
P
P
(d2 )]
(d1 ) ∧ IB
(b2 ] ∧ [IB
(b1 ) ∧ IB
≤ [IB
2
2
1
1
P
P
P
P
(d2 )]
(b2 ) ∧ IB
(d1 )] ∧ [IB
(b1 ) ∧ IB
= [IB
2
1
2
1
P
P
= I(B
(b1 d1 ) ∧ I(B
(b2 d2 ),
1 ⊠B2 )
1 ⊠B2 )

Bipolar Neutrosophic Graph Structures

71

N
N
N
(d1 d2 )
I(B
((b1 d1 )(b2 d2 )) = IB
(b1 b2 ) ∨ IB
2k
1k
1k ⊠B2k )
N
N
N
N
(d2 )]
(d1 ) ∨ IB
(b2 ] ∨ [IB
(b1 ) ∨ IB
≥ [IB
2
2
1
1
N
N
N
N
(d2 )]
(b2 ) ∨ IB
(d1 )] ∨ [IB
(b1 ) ∨ IB
= [IB
2
1
2
1
N
N
(b2 d2 ),
= I(B
(b1 d1 ) ∨ I(B
1 ⊠B2 )
1 ⊠B2 )
P
((b1 d1 )(b2 d2 )) = FBP1k (b1 b2 ) ∨ FBP2k (d1 d2 )
F(B
1k ⊠B2k )

≤ [FBP1 (b1 ) ∨ FBP1 (b2 ] ∨ [FBP2 (d1 ) ∨ FBP2 (d2 )]
= [FBP1 (b1 ) ∨ FBP2 (d1 )] ∨ [FBP1 (b2 ) ∨ FBP2 (d2 )]
P
P
(b2 d2 ),
(b1 d1 ) ∨ F(B
= F(B
1 ⊠B2 )
1 ⊠B2 )
N
F(B
((b1 d1 )(b2 d2 )) = FBN1k (b1 b2 ) ∧ FBN2k (d1 d2 )
1k ⊠B2k )

≥ [FBN1 (b1 ) ∧ FBN1 (b2 ] ∧ [FBN2 (d1 ) ∧ FBN2 (d2 )]
= [FBN1 (b1 ) ∧ FBN2 (d1 )] ∧ [FBN1 (b2 ) ∧ FBN2 (d2 )]
N
N
= F(B
(b1 d1 ) ∧ F(B
(b2 d2 ),
1 ⊠B2 )
1 ⊠B2 )

b1 d1 , b2 d2 ∈ V1 ⊠ V2 .
All cases hold ∀ k ∈ {1, 2, . . . , m}.
ˇ b1 = (B1 , B11 , B12 , . . . , B1m ) and G
ˇ b2 = (B2 , B21 , B22 , . . . , B2m )
Definition 2.25. Let G
ˇ
ˇ
be BSVNGSs. Union of Gb1 and Gb2 , denoted by
ˇ b1 ∪ G
ˇ b2 = (B1 ∪ B2 , B11 ∪ B21 , B12 ∪ B22 , . . . , B1m ∪ B2m ),
G
is defined as:

P
 T(B
(b) = (TBP1 ∪ TBP2 )(b) = TBP1 (b) ∨ TBP2 (b)

1 ∪B2 )
P
P
P
P
P
I(B1 ∪B2 ) (b) = (IB
∪ IB
)(b) = (IB
(b) + IB
(b))/2
(i)
1
2
1
2

P
P
P
P
P
 F
(b)
=
(F
∪
F
)(b)
=
F
(b)
∧
F
B1
B2
B1
B2 (b)
 (B1 ∪B2 )
N
N
N
N
N

 T(B1 ∪B2 ) (b) = (TB1 ∪ TB2 )(b) = TB1 (b) ∧ TB2 (b)
N
N
N
N
N
I(B1 ∪B2 ) (b) = (IB1 ∪ IB2 )(b) = (IB1 (b) + IB2 (b))/2
(ii)

N
N
N
N
 FN
(B1 ∪B2 ) (b) = (FB1 ∪ FB2 )(b) = FB1 (b) ∨ FB2 (b)
for
 all b ∈ V1 ∪ V2 ,
P
P
P
P
P

 T(B1k ∪B2k ) (bd) = (TB1k ∪ TB2k )(bd) = TB1k (bd) ∨ TB2k (bd)
P
P
P
P
P
I(B1k ∪B2k ) (bd) = (IB1k ∪ IB2k )(bd) = (IB1k (bd) + IB2k (bd))/2
(iii)

 FP
(bd) = (FBP1k ∪ FBP2k )(bd) = FBP1k (bd) ∧ FBP2k (bd)
 (B1k ∪B2k )
N
N
N
N
N

 T(B1k ∪B2k ) (bd) = (TB1k ∪ TB2k )(bd) = TB1k (bd) ∧ TB2k (bd)
N
N
N
N
N
I(B1k ∪B2k ) (bd) = (IB1k ∪ IB2k )(bd) = (IB1k (bd) + IB2k (bd))/2
(iv)

N
N
N
N
 FN
(B1k ∪B2k ) (bd) = (FB1k ∪ FB2k )(bd) = FB1k (bd) ∨ FB2k (bd)
for all (bd) ∈ V1k ∪ V2k .

M. Akram and M. Sitara

72

B12 ∪ B22 (0.3, 0.1, 0.5, −0.3, −0.1, −0.5)
B11 ∪ B21 (0.5, 0.05, 0.8, −0.5, −0.05, −0.8)

b4 (0.5, 0.1, 0.6, −0.5, −0.1, −0.6)

b

b

b2 (0.5, 0.1, 0.8, −0.5, −0.1, −0.8)

d3 (0.5, 0.15, 0.5, −0.5, −0.15, −0.5)

b

d1 (0.2, 0.05, 0.3, −0.2, −0.05, −0.3)
b

B11 ∪ B21 (0.2, 0.05, 0.4, −0.2, −0.05, −0.4)
B12 ∪ B22 (0.4, 0.1, 0.6, −0.4, −0.1, −0.6)

b
b

b

b1 (0.5, 0.05, 0.6, −0.5, −0.05, −0.6)

d2 (0.3, 0.1, 0.4, −0.3, −0.1, −0.4)

b3 (0.4, 0.1, 0.4, −0.4, −0.1, −0.4)

ˇ b1 ∪ G
ˇ b2
Figure 11. G
ˇ b1 and G
ˇ b2 shown in Fig. 7 is defined
Example 2.26. Union of two BSVNGSs G
ˇ
ˇ
as Gb1 ∪ Gb2 = {B1 ∪ B2 , B11 ∪ B21 , B12 ∪ B22 } and is depicted in Fig. 11.
ˇ b1 ∪ G
ˇ b2 = (B1 ∪ B2 , B11 ∪ B21 , B12 ∪ B22 , . . . , B1m ∪ B2m )
Theorem 2.27. Union G
of two BSVNGSs of the GSs Gˇ1 and Gˇ2 is BSVNGS of Gˇ1 ∪ Gˇ2 .
proof.Let b1 b2 ∈ V1k ∪ V2k . Two cases arise:
Case 1.: For b1 , b2 ∈ V1 , by definition 2.25
P
P
P
TBP2 (b1 ) = TBP2 (b2 ) = TBP2k (b1 b2 ) = 0, IB
(b1 ) = IB
(b2 ) = IB
(b1 b2 ) = 0,
2
2
2k
P
P
P
FB2 (b1 ) = FB2 (b2 ) = FB2k (b1 b2 ) = 1,
N
N
N
TBN2 (b1 ) = TBN2 (b2 ) = TBN2k (b1 b2 ) = 0, IB
(b1 ) = IB
(b2 ) = IB
(b1 b2 ) = 0,
2
2
2k
N
N
N
FB2 (b1 ) = FB2 (b2 ) = FB2k (b1 b2 ) = −1, so
P
T(B
(b1 b2 ) = TBP1k (b1 b2 ) ∨ TBP2k (b1 b2 )
1k ∪B2k )

= TBP1k (b1 b2 ) ∨ 0
≤ [TBP1 (b1 ) ∧ TBP1 (b2 )] ∨ 0
= [TBP1 (b1 ) ∨ 0] ∧ [TBP1 (b2 ) ∨ 0]
= [TBP1 (b1 ) ∨ TBP2 (b1 )] ∧ [TBP1 (b2 ) ∨ TBP2 (b2 )]
P
P
(b2 ),
(b1 ) ∧ T(B
= T(B
1 ∪B2 )
1 ∪B2 )
N
T(B
(b1 b2 ) = TBN1k (b1 b2 ) ∧ TBN2k (b1 b2 )
1k ∪B2k )

= TBN1k (b1 b2 ) ∧ 0
≥ [TBN1 (b1 ) ∨ TBN1 (b2 )] ∧ 0
= [TBN1 (b1 ) ∧ 0] ∨ [TBN1 (b2 ) ∧ 0]
= [TBN1 (b1 ) ∧ TBN2 (b1 )] ∨ [TBN1 (b2 ) ∧ TBN2 (b2 )]
N
N
= T(B
(b1 ) ∨ T(B
(b2 ),
1 ∪B2 )
1 ∪B2 )

Bipolar Neutrosophic Graph Structures
P
F(B
(b1 b2 ) = FBP1k (b1 b2 ) ∧ FBP2k (b1 b2 )
1k ∪B2k )

= FBP1k (b1 b2 ) ∧ 1
≤ [FBP1 (b1 ) ∨ FBP1 (b2 )] ∧ 1
= [FBP1 (b1 ) ∧ 1] ∨ [FBP1 (b2 ) ∧ 1]
= [FBP1 (b1 ) ∧ FBP2 (b1 )] ∨ [FBP1 (b2 ) ∧ FBP2 (b2 )]
P
P
(b2 ),
(b1 ) ∨ F(B
= F(B
1 ∪B2 )
1 ∪B2 )

N
F(B
(b1 b2 ) = FBN1k (b1 b2 ) ∨ FBN2k (b1 b2 )
1k ∪B2k )

= FBN1k (b1 b2 ) ∨ −1
≥ [FBN1 (b1 ) ∧ FBN1 (b2 )] ∨ −1
= [FBN1 (b1 ) ∨ −1] ∧ [FBN1 (b2 ) ∨ −1]
= [FBN1 (b1 ) ∨ FBN2 (b1 )] ∧ [FBN1 (b2 ) ∨ FBN2 (b2 )]
N
N
(b2 ),
(b1 ) ∧ F(B
= F(B
1 ∪B2 )
1 ∪B2 )

P
P
IB
(b1 b2 ) + IB
(b1 b2 )
1k
2k
2
P
IB
(b
b
)
+
0
1
2
= 1k
2
P
P
[IB
(b
) ∧ IB
(b2 )] + 0
1
1
1
≤
2
I P (b2 )
I P (b1 )
+ 0] ∧ [ B1
+ 0]
= [ B1
2
2
P
P
P
(b2 ) + IB
(b2 )]
[I P (b1 ) + IB
(b1 )] [IB
1
2
2
∧
= B1
2
2
P
P
= I(B
(b
)
∧
I
(b
),
1
2
(B1 ∪B2 )
1 ∪B2 )

P
(b1 b2 ) =
I(B
1k ∪B2k )

73

74

M. Akram and M. Sitara
N
N
IB
(b1 b2 ) + IB
(b1 b2 )
1k
2k
2
N
IB
(b
b
)
+
0
1
2
= 1k
2
N
[I N (b1 ) ∨ IB
(b2 )] + 0
1
≥ B1
2
N
IB
(b
)
I N (b2 )
1
=[ 1
+ 0] ∨ [ B1
+ 0]
2
2
N
N
N
(b2 ) + IB
(b2 )]
[I N (b1 ) + IB
(b1 )] [IB
1
2
2
∨
= B1
2
2
N
N
(b2 ),
(b1 ) ∨ I(B
= I(B
1 ∪B2 )
1 ∪B2 )

N
I(B
(b1 b2 ) =
1k ∪B2k )

for b1 , b2 ∈ V1 ∪ V2 .
Case 2.: For b1 , b2 ∈ V2 , by definition 2.25
P
P
P
TBP1 (b1 ) = TBP1 (b2 ) = TBP1k (b1 b2 ) = 0, IB
(b1 ) = IB
(b2 ) = IB
(b1 b2 ) = 0,
1
1
1k
P
P
P
FB1 (b1 ) = FB2 (b2 ) = FB1k (b1 b2 ) = 1,
N
N
N
TBN1 (b1 ) = TBN1 (b2 ) = TBN1k (b1 b2 ) = 0, IB
(b1 ) = IB
(b2 ) = IB
(b1 b2 ) = 0,
1
1
1k
N
N
N
FB1 (b1 ) = FB2 (b2 ) = FB1k (b1 b2 ) = −1, so

P
(b1 b2 ) = TBP1k (b1 b2 ) ∨ TBP2k (b1 b2 )
T(B
1k ∪B2k )

= TBP2k (b1 b2 ) ∨ 0
≤ [TBP2 (b1 ) ∧ TBP2 (b2 )] ∨ 0
= [TBP2 (b1 ) ∨ 0] ∧ [TBP2 (b2 ) ∨ 0]
= [TBP2 (b1 ) ∨ TBP1 (b1 )] ∧ [TBP2 (b2 ) ∨ TBP1 (b2 )]
P
P
(b2 ),
(b1 ) ∧ T(B
= T(B
1 ∪B2 )
1 ∪B2 )

N
T(B
(b1 b2 ) = TBN1k (b1 b2 ) ∧ TBN2k (b1 b2 )
1k ∪B2k )

= TBN2k (b1 b2 ) ∧ 0
≥ [TBN2 (b1 ) ∨ TBN2 (b2 )] ∧ 0
= [TBN2 (b1 ) ∧ 0] ∨ [TBN2 (b2 ) ∧ 0]
= [TBN2 (b1 ) ∧ TBN1 (b1 )] ∨ [TBN2 (b2 ) ∧ TBN1 (b2 )]
N
N
= T(B
(b1 ) ∨ T(B
(b2 ),
1 ∪B2 )
1 ∪B2 )

Bipolar Neutrosophic Graph Structures
P
F(B
(b1 b2 ) = FBP1k (b1 b2 ) ∧ FBP2k (b1 b2 )
1k ∪B2k )

= FBP2k (b1 b2 ) ∧ (1)
≤ [FBP2 (b1 ) ∨ FBP2 (b2 )] ∧ (1)
= [FBP2 (b1 ) ∧ (1)] ∨ [FBP2 (b2 ) ∧ (1)]
= [FBP2 (b1 ) ∧ FBP1 (b1 )] ∨ [FBP2 (b2 ) ∧ FBP1 (b2 )]
P
P
(b2 ),
(b1 ) ∨ F(B
= F(B
1 ∪B2 )
1 ∪B2 )

N
F(B
(b1 b2 ) = FBN1k (b1 b2 ) ∨ FBN2k (b1 b2 )
1k ∪B2k )

= FBN2k (b1 b2 ) ∨ (−1)
≥ [FBN2 (b1 ) ∧ FBN2 (b2 )] ∨ (−1)
= [FBN2 (b1 ) ∨ (−1)] ∧ [FBN2 (b2 ) ∨ (−1)]
= [FBN2 (b1 ) ∨ FBN1 (b1 )] ∧ [FBN2 (b2 ) ∨ FBN1 (b2 )]
N
N
(b2 ),
(b1 ) ∧ F(B
= F(B
1 ∪B2 )
1 ∪B2 )

P
P
IB
(b1 b2 ) + IB
(b1 b2 )
1k
2k
2
P
IB
(b
b
)
+
0
1
2
= 2k
2
P
P
[IB
(b
) ∧ IB
(b2 )] + 0
1
2
2
≤
2
I P (b2 )
I P (b1 )
+ 0] ∧ [ B2
+ 0]
= [ B2
2
2
P
P
P
(b2 ) + IB
(b2 )]
[I P (b1 ) + IB
(b1 )] [IB
2
1
1
∧
= B2
2
2
P
P
= I(B
(b
)
∧
I
(b
),
1
2
(B1 ∪B2 )
1 ∪B2 )

P
(b1 b2 ) =
I(B
1k ∪B2k )

75

M. Akram and M. Sitara

76

N
N
IB
(b1 b2 ) + IB
(b1 b2 )
1k
2k
2
N
IB
(b
b
)
+
0
1
2
= 2k
2
N
[I N (b1 ) ∨ IB
(b2 )] + 0
2
≥ B2
2
N
IB
(b
)
I N (b2 )
1
=[ 2
+ 0] ∨ [ B2
+ 0]
2
2
N
N
N
(b2 ) + IB
(b2 )]
[I N (b1 ) + IB
(b1 )] [IB
2
1
1
∨
= B2
2
2
N
N
(b2 ),
(b1 ) ∨ I(B
= I(B
1 ∪B2 )
1 ∪B2 )

N
I(B
(b1 b2 ) =
1k ∪B2k )

for b1 , b2 ∈ V1 ∪ V2 .
Both cases hold ∀ k ∈ {1, 2, . . . , m}. This completes the proof.
Theorem 2.28. Let Gˇs = (V1 ∪ V2 , V11 ∪ V21 , V12 ∪ V22 , . . . , V1m ∪ V2m ) be union
of GSs Gˇs1 = (V1 , V11 , V12 , . . . , V1m ) and Gˇs2 = (V2 , V21 , V22 , . . . , V2m ). Then every
ˇ bn = (B, B1 , B2 , . . . , Bm ) of Gˇs is union of two BSVNGSs G
ˇ b1 and G
ˇ b2
BSVNGS G
ˇ
ˇ
of GSs Gs1 and Gs2 , respectively.
proofFirstly, we define B1 , B2 , B1k and B2k for k ∈ {1, 2, . . . , m} as:
P
P
TBP1 (b) = TBP (b), IB
(b) = IB
(b), FBP1 (b) = FBP (b),
1
N
N
N
N
TB1 (b) = TB (b), IB1 (b) = IB (b), FBN1 (b) = FBN (b), if b ∈ V1 .
P
P
TBP2 (b) = TBP (b), IB
(b) = IB
(b), FBP2 (b) = FBP (b),
2
N
N
N
N
TB2 (b) = TB (b), IB2 (b) = IB (b), FBN2 (b) = FBN (b), if b ∈ V2 .
P
P
TBP1k (b1 b2 ) = TBPk (b1 b2 ), IB
(b1 b2 ) = IB
(b1 b2 ), FBP1k (b1 b2 ) = FBPk (b1 b2 ), TBN1k (b1 b2 ) =
1k
k
N
N
TBNk (b1 b2 ), IB
(b1 b2 ) = IB
(b1 b2 ), FBN1k (b1 b2 ) = FBNk (b1 b2 ), if b1 b2 ∈ V1k .
1k
k
P
P
P
P
TB2k (b1 b2 ) = TBk (b1 b2 ), IB2k (b1 b2 ) = IB
(b1 b2 ), FBP2k (b1 b2 ) = FBPk (b1 b2 ), TBN2k (b1 b2 ) =
k
N
N
N
N
TBk (b1 b2 ), IB2k (b1 b2 ) = IBk (b1 b2 ), FB2k (b1 b2 ) = FBNk (b1 b2 ), if b1 b2 ∈ V2k .

Then B = B1 ∪ B2 and Bk = B1k ∪ B2k , k ∈ {1, 2, . . . , m}. Now for b1 b2 ∈ Vtk ,
t = 1, 2, k ∈ {1, 2, . . . , m}:
P
TBPtk (b1 b2 ) = TBPk (b1 b2 ) ≤ TBP (b1 ) ∧ TBP (b2 ) = TBPt (b1 ) ∧ TBPt (b2 ), IB
(b1 b2 ) =
tk
P
P
P
P
P
P
P
IB
(b
b
)
≤
I
(b
)∧I
(b
)
=
I
(b
)∧I
(b
),
F
(b
b
)
=
F
(b
b
)
≤
FBP (b1 )∨
1 2
B 1
B 2
Bt 1
Bt 2
Btk 1 2
Bk 1 2
k
P
P
P
FB (b2 ) = FBt (b1 ) ∨ FBt (b2 ),
N
TBNtk (b1 b2 ) = TBNk (b1 b2 ) ≥ TBN (b1 ) ∨ TBN (b2 ) = TBNt (b1 ) ∨ TBNt (b2 ), IB
(b1 b2 ) =
tk
N
N
N
N
N
N
IBk (b1 b2 ) ≥ IB (b1 ) ∨ IB (b2 ) = IBt (b1 ) ∨ IBt (b2 ), FBtk (b1 b2 ) = FBNk (b1 b2 ) ≥
FBN (b1 ) ∧ FBN (b2 ) = FBNt (b1 ) ∧ FBNt (b2 ), i.e.,
ˇ bl = (Bl , Bl1 , Bl2 , . . . , Blm ) is a BSVNGS of G
ˇ t , t = 1, 2. Thus G
ˇ bn = (B, B1 , B2 , . . . , Bm ),
G
ˇ
ˇ
ˇ
ˇ
ˇ b2 .
a BSVNGS of Gs = Gs1 ∪ Gs2 , is the union of two BSVNGSs Gb1 and G

Bipolar Neutrosophic Graph Structures

77

ˇ b1 = (B1 , B11 , B12 , . . . , B1m ) and G
ˇ b2 = (B2 , B21 , B22 , . . . , B2m )
Definition 2.29. Let G
ˇ b1 and G
ˇ b2 , denoted by
be BSVNGSs and let V1 ∩ V2 = ∅. Join of G
ˇ b1 + G
ˇ b2 = (B1 + B2 , B11 + B21 , B12 + B22 , . . . , B1m + B2m ),
G
is defined
 as:
P
P

 T(B1 +B2 ) (b) = T(B1 ∪B2 ) (b)
P
P
I(B1 +B2 ) (b) = I(B1 ∪B2 ) (b)
(i)

P
 FP
(b) = F(B
(b)
1 ∪B2 )
 (B1 +B2 )
N
N

 T(B1 +B2 ) (b) = T(B1 ∪B2 ) (b)
N
N
I(B1 +B2 ) (b) = I(B1 ∪B2 ) (b)
(ii)

N
 FN
(B1 +B2 ) (b) = F(B1 ∪B2 ) (b)
for
 all b ∈ V1 ∪ V2 ,
P
P

 T(B1k +B2k ) (bd) = T(B1k ∪B2k ) (bd)
P
P
I(B1k +B2k ) (bd) = I(B1k ∪B2k ) (bd)
(iii)

P
 FP
(bd) = F(B
(bd)
1k ∪B2k )
 (B1k +B2k )
N
N

 T(B1k +B2k ) (bd) = T(B1k ∪B2k ) (bd)
N
N
I(B1k +B2k ) (bd) = I(B1k ∪B2k ) (bd)
(iv)

N
 FN
(B1k +B2k ) (bd) = F(B1k ∪B2k ) (bd)
for
 all (bd) ∈ V1k ∪ V2k ,
P
P
P
P
P

 T(B1k +B2k ) (bd) = (TB1k + TB2k )(bd) = TB1 (b) ∧ TB2 (d)
P
P
P
P
P
I(B
(bd) = (IB
+ IB
)(bd) = IB
(b) ∧ IB
(d)
(v)
1
2
1k
2k
1k +B2k )

P
P
P
 FP
(bd) = (FB1k + FB2k )(bd) = FB1 (b) ∨ FBP2 (d)
 (B1k +B2k )
N
N
N
N
N

 T(B1k +B2k ) (bd) = (TB1k + TB2k )(bd) = TB1 (b) ∨ TB2 (d)
N
N
N
N
N
I(B1k +B2k ) (bd) = (IB1k + IB2k )(bd) = IB1 (b) ∨ IB2 (d)
(vi)

N
N
N
N
 FN
(B1k +B2k ) (bd) = (FB1k + FB2k )(bd) = FB1 (b) ∧ FB2 (d)
for all b ∈ V1 , d ∈ V2 .
ˇ b1 and G
ˇ b2 shown in Fig. 7 is defined as
Example 2.30. Join of two BSVNGSs G
ˇ
ˇ
Gb1 + Gb2 = {B1 + B2 , B11 + B21 , B12 + B22 } and is depicted in Fig. 12.
ˇ b1 + G
ˇ b2 = (B1 + B2 , B11 + B21 , B12 + B22 , . . . , B1m + B2m )
Theorem 2.31. Join G
of two BSVNGSs of the GSs Gˇ1 and Gˇ2 is BSVNGS of Gˇ1 + Gˇ2 .
3. CONCLUDING REMARKS
Bipolar fuzzy graph theory has numerous applications in various fields of
science and technology including, artificial intelligence, operations research and decision making. A bipolar neutrosophic graph constitutes a generalization of the
notion bipolar fuzzy graph. In this research paper, We have introduced the idea
of bipolar single-valued neutrosophic graph structure and discussed many relevant
notions. We also discussed a worthwhile application of bipolar single-valued neutrosophic graph structure in decision-making. In future, we aim to generalize our notions to (1) BSVN hypergraph structures, (2) BSVN vague hypergraph structures,

.4)
−0

ˇ b1 + G
ˇ b2
Figure 12. G
(3) BSVN interval-valued hypergraph structures, and(4) BSVN rough hypergraph
structures.
Acknowledgement. The authors are highly thankful to Executive Editor and
the referees for their valuable comments and suggestions.

References
[1] Akram, M., ”Bipolar fuzzy graphs”, Information Sciences”, 181:24 (2011), 5548-5564.
[2] Akram, M., ”Single-valued neutrosophic planar graphs”, International Journal of Algebra
and Statistics, 5(2) (2016), 157-167.
[3] Akram, M. and Sarwar, M., ”Novel multiple criteria decision making methods based on
bipolar neutrosophic sets and bipolar neutrosophic graphs”, viXra:1701.0190, 2015.
[4] Akram, M. and Akmal, R., ”Application of bipolar fuzzy sets in graph structures”, Applied
Computational Intelligence and Soft Computing, (2016), 13 pages.
[5] Akram, M. and Shahzadi, S., ”Neutrosophic soft graphs with application”, Journal of Intelligent & Fuzzy Systems, 32 (2017), 841-858.
[6] Akram, M. and Shahzadi, S., ”Representation of graphs using intuitionistic neutrosophic soft
sets”, Journal of Mathematical Analysis, 7:6 (2016), 31-53.
[7] Atanassov, K., ”Intuitionistic fuzzy sets”, Fuzzy Sets and Systems, 20:1 (1986), 87-96.
[8] Bhattacharya, P., ”Some remarks on fuzzy graphs”, Pattern Recognition Letters, 6:5 (1987),
297-302.
[9] Deli, I., Ali, M. and Smarandache, F., Bipolar neutrosophic sets and their application based
on multi-criteria decision making problems, In Advanced Mechatronic Systems (ICAMechS),
International Conference IEEE, 249-254.

d1 (0.2, 0.05, 0.3, −0.2, −0.05, −0.3)

−0
.8)

−0
.1,

−0
.3,

0.8
,

.1,
−0

0.1
,

.3,
−0

(0.
3,

,
0.4

d3 (0.5, 0.15, 0.5, −0.5, −0.15, −0.5)

,
0.1

B11 + B21 (0.5, 0.05, 0.8, −0.5, −0.05, −0.8)

3,
(0.

B
) d (0.3, 0.1, 0.4, −0.3, −0.1, −0.4) 11 +
2
B2
0.5
−
b
,
1 (0
(
.2,
0.1
.6) 0.3,
0.0
,−
)
0
3
6
.
0
5, 0
.
b4
0
−
.1,
0
,
−
(0
.3,
5
,
−
0
5
0
,
.
.
.6,
.
−0
5,
5
0
0
,
0
.2,
1
.
0.1
−0
.
−
0
0
,
−0
,
3
,
−
.
3
.
.
3
0
,
.05
0
0
,
.
5
(
6
.
−
−
,−
,−
0
,
0
B 22
6
.
−
0.3
.
0.5
1,
,
+
0
6
)
−0
,
.
2
,
5
0
1
−
,
0
B
.
.
0
6
5
0
.1,
) (0
,
)
.0
3
0
6
.
b
−0
2,
0. b (0.
0.0
.5,
.6)
,−
(0
5,
b1
0.6
.05
0
,
−
−
0.2
.5,
,−
−0
0.0
.6,
(
)
0
0
.2, 0.0
5,
0.6
,
−
,
5
5
−0
, 0.6, −
0.1
0
.
−
,
0
0.2, −
.6)
0.5
0.05,
.6, −
0
.5b ,
,
.1
b
0
−
0.6)
(
.5, 0
0
(
(0
.4)
0
.5,
−
0.1
5,
)
(0.4, 0.1
.8
0
0
.
−
,0
,
, 0.5, −
.8,
−0.05
−0
0.4, −0
−0.2,
−0
.2,
.1, −0.5
5, 0.8,
−0
.5,
)
.2, 0.0
,
(0
4
.
−0
,0
.1,
.05
−0 b
b
,0
2
.
.8)
(0
b3 (0.4, 0.1, 0.4, −0.4, −0.1, −0.4)
b2 (0.5, 0.1, 0.8, −0.5, −0.1, −0.8)

B12 + B22 (0.4, 0.1, 0.6, −0.4, −0.1, −0.6)

M. Akram and M. Sitara

78

Bipolar Neutrosophic Graph Structures

79

[10] Dhavaseelan, R., Vikramaprasad, R. and Krishnaraj, V., ”Certain types of neutrosophic
graphs”, Int. Jr. of Mathematical Sciences and Applications, 5:) (2015), 333-339.
[11] Dinesh, T., A study on graph structures, incidence algebras and their fuzzy analogues[Ph.D.thesis], Kannur University, Kannur, India, (2011).
[12] Dinesh, T. and Ramakrishnan, T.V., ”On generalised fuzzy graph structures”, Applied Mathematical Sciences, 5:4 (2011), 173-180.
[13] Kauffman, A., Introduction a la Theorie des Sous-emsembles Flous, Masson et Cie, Vol. 1
(1973).
[14] Mordeson, J.N. and Chang-Shyh, P., ”Operations on fuzzy graphs”, Information Sciences,
79 (1994), 159-170.
[15] Peng, J.J., Wang, J.Q., Zhang, H.Y. and Chen, X.H, ”An outranking approach for multicriteria decision-making problems with simplified neutrosophic sets”, Applied Soft Computing, 25 (2014), 336-346.
[16] Rosenfeld, A., ”Fuzzy graphs”, In Fuzzy Sets and their Applications( L.A.Zadeh, K.S.Fu,
M.Shimura, Eds.), Academic Press, New York, (1975), 77-95.
[17] Sampathkumar, E., ”Generalized graph structures”, Bulletin of Kerala Mathematics Association, 3:2 (2006), 65-123.
[18] Shah, N. and Hussain, A., ”Neutrosophic soft graphs”, Neutrosophic Set and Systems, 11
(2016), 31-44.
[19] Smarandache, F., Neutrosophy Neutrosophic Probability, Set, and Logic, Amer Res Press,
Rehoboth, USA, (1998).
[20] Sunitha, M.S. and Vijayakumar, A., ”Complement of a fuzzy graph”, Indian Journal of Pure
and Applied Mathematics, 33:) (2002), 1451-1464.
[21] Wang, H., Smarandache, F., Zhang, Y.Q. and Sunderraman, R., ”Single valued neutrosophic
sets”, Multispace and Multistructure, 4 (2010), 410-413.
[22] Ye, J., ”Single-valued neutrosophic minimum spanning tree and its clustering method”, Journal of Intelligent Systems, 23:3 (2014), 311-324.
[23] Zadeh, L.A., ”Fuzzy sets”, Information and control, 8:3 (1965), 338-353.
[24] Zadeh, L.A., ”Similarity relations and fuzzy orderings”, Information Sciences, 3:2 (1971),
177-200.
[25] Zhang, W.-R., ”Bipolar fuzzy sets and relations: a computational framework for cognitive
modeling and multiagent decision analysis”, In Fuzzy Information Processing Society Biannual Conference, 1994. Industrial Fuzzy Control and Intelligent systems Conference, and
the NASA Joint Technology Workshop on Neural Networks and Fuzzy Logic, IEEE, (1994),
305-309.

80

M. Akram and M. Sitara