BIPOLAR NEUTROSOPHIC GRAPH STRUCTURES

Vol. 23, No. 1 (2017), pp. 55–80.

BIPOLAR NEUTROSOPHIC GRAPH STRUCTURES

Muhammad Akram¹ and Muzzamal Sitara²

1Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan,

[email protected]

2Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan,

[email protected]

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 con- struction 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 neu- trosofik bipolar bernilai tunggal. Kami mengkaji ide-ide tertentu dari struktur graf neutrosofik bipolar bernilai tunggal berserta contoh-contohnya. Kami menya- jikan 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 sys- tems 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 re- ducing 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

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 de- scribed 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 re- lated 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 appli- cations. 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: truth- membership, 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, T^N, I^N, F^N : V →]0⁻, 1⁺[ and there is no restriction on the sum of T^N(v), I^N(v) and F^N(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, T^N(v), I^N(v), F^N(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 .

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

B = {(v, TB^P(v), IB^P(v), FB^P(v), TB^N(v), IB^N(v), FB^N(v)) : v ∈ V }

where, TB^P, IB^P, FB^P : V → [0, 1] and TB^N, IB^N, FB^N : V → [−1, 0]. The positive values TB^P(v), IB^P(v), FB^P(v) denote the truth, indeterminacy and falsity membership values of an element v ∈ V , whereas negative values TB^N(v), IB^N(v), FB^N(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

TR^P(bd) ≤ TB^P(b) ∧ TB^P(d), IR^P(bd) ≤ IB^P(b) ∧ IB^P(d), FR^P(bd) ≤ FB^P(b) ∨ FB^P(d), TR^N(bd) ≥ TB^N(b) ∨ TB^N(d), IR^N(bd) ≥ IB^N(b) ∨ IB^N(d), FR^N(bd) ≥ FB^N(b) ∧ FB^N(d),

for all b, d ∈ V.

We now define bipolar single-valued neutrosophic graph structure.

Definition 2.5. ˇG^bn= (B, B1, B2, . . . , B^m) is called bipolar single-valued neutro- sophic graph structure(BSVNGS) of graph structure ˇG^s = (V, V1, V2, . . . , V^m) if B =< b, T^P(b), I^P(b), F^P(b), T^N(b), I^N(b), F^N(b) > and

B^k =< (b, d), Tk^P(b, d), Ik^P(b, d), Fk^P(b, d), Tk^N(b, d), Ik^N(b, d), Fk^N(b, d) > are bipolar single-valued neutrosophic(BSVN) sets on V and V^k, respectively, such that

Tk^P(b, d) ≤ min{T^P(b), T^P(d)}, Ik^P(b, d) ≤ min{I^P(b), I^P(d)}, Fk^P(b, d) ≤ max{F^P(b), F^P(d)}, Tk^N(b, d) ≥ max{T^N(b), T^N(d)},

Ik^N(b, d) ≥ max{I^N(b), I^N(d)}, Fk^N(b, d) ≥ min{F^N(b), F^N(d)}, for all b, d ∈ V. Note that 0 ≤ Tk^P(b, d) + Ik^P(b, d) + Fk^P(b, d) ≤ 3,

−3 ≤ Tk^N(b, d) + Ik^N(b, d) + Fk^N(b, d) ≤ 0 for all (b, d) ∈ V^k.

Example 2.6. Consider graph structure(GSR) ˇGs = (V, V1, V2) such that V = {b1, b2, b3, b4}, V1= {b1b3, b1b2, b3b4}, V2= {b1b4, b2b3}. By defining bipolar single- valued neutrosophic sets B, B1and B2on V , V1 and V2, respectively, we can draw a bipolar SVNGS as depicted in Fig. 1.

Definition 2.7. Let ˇG^bn = (B, B1, B2, . . . , B^m) be a BSVNGS of GS ˇG^s. If Hˇ^bn= (B^′, B^′₁, B₂^′, . . . , B^′m) is a BSVNGS of ˇG^s such that

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) ,

b

b b

b

B₁ (0.2,0

.3, 0.4, −0.2, −0.3, −0.4)

b4(0.3, 0.2, 0.3, −0.3, −0.2, −0.3)

b3(0.3,0.4,0.3,−0.3,−0.4,−0.3) b2(0.2, 0.2, 0.3, −0.2, −0.2, −0.3)

b1(0.2, 0.3, 0.4, −0.2, −0.3, −0.4) B

2(0 .2,0

.2,0 .4,−

0.2,−

0.2,

− 0.4)

B¹(0.3,0.2, 0.3,−0.3,−0.2,−0.3)

B²(0.2, 0.2,0.3, −0.2, −0.2, −0.3) B1(0.2,0.2,0.4,−0.2,−0.2,−0.4)

Figure 1. A bipolar single-valued neutrosophic graph structure

b

b b

b

B₁^′

(0.1,0.2, 0.5, −0.1, −0.2 , −0.5) b4(0.2, 0.1, 0.4, −0.2, −0.1, −0.4)

b3(0.2,0.3,0.4,−0.2,−0.3,−0.4) b2(0.1, 0.1, 0.4, −0.1, −0.1, −0.4)

b1(0.1, 0.2, 0.5, −0.1, −0.2, −0.5) B

′ 2(0 .1,0

.1,0 .5,−

0.1,− 0.1,−

0.5)

B^′1(0.2,0.1, 0.4,−0.2,−0.1,−0.4)

B^′2(0.1, 0.1,0.4, −0.1, −0.1, −0.4) B′ 1(0.1,0.1,0.5,−0.1,−0.1,−0.5)

Figure 2. A BSVN subgraph structure

Tk^′P(b, d) ≤ Tk^P(b, d), Ik^′P(b, d) ≤ Ik^P(b, d), Fk^′P(b, d) ≥ Fk^P(b, d), Tk^′N(b, d) ≥ Tk^N(b, d), Ik^′N(b, d) ≥ Ik^N(b, d), Fk^′N(b, d) ≤ Fk^N(b, d),

∀ b ∈ V and (b, d) ∈ V^k, k = 1, 2, . . . , m.

Then ˇH^bn is named as a bipolar single-valued neutrosophic(BSVN) subgraph struc- ture of BSVNGS ˇGbn.

Example 2.8. Consider a BSVNGS ˇH^bn = (B^′, B₁^′, B₂^′) of GS ˇG^s = (V, V1, V2) as depicted in Fig. 2. Routine calculations indicate that ˇH^bnis BSVN subgraph- structure of BSVNGS ˇG^bn.

Definition 2.9. A BSVNGS ˇH^bn= (B^′, B₁^′, B^′₂, . . . , Bm^′ ) is called a BSVN induced subgraph-structure of BSVNGS ˇGbnby Q ⊆ V if

b

b

b

b³(0.3, 0.4, 0.3, −0.3,−0.4,−0.3) b2(0.2,0.2,0.3,−0.2,−0.2,−0.3)

b₁ (0.2, 0.3

, 0.4, − 0.2, −

0.3, − 0.4) B₁^′(0.2, 0.3, 0.4, −0.2, −0.3, −0.4) B′

2(0.2 , 0.2,0

.3,− 0.2, −

0.2, − 0.3) B^′1(0.2,0.2,0.4,−0.2, −0.2, −0.4)

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) = Tk^P(b, d), Ik^′P(b, d) = Ik^P(b, d), Fk^′P(b, d) = Fk^P(b, d), Tk^′N(b, d) = Tk^N(b, d), Ik^′N(b, d) = Ik^N(b, d), Fk^′N(b, d) = Fk^N(b, d),

∀ b, d ∈ Q, k = 1, 2, . . . , m.

Example 2.10. A BSVNGS depicted in Fig.3 is a BSVN induced subgraph- structure of BSVNGS represented in Fig. 1.

Definition 2.11. A BSVNGS ˇH^bn= (B^′, B₁^′, B^′₂, . . . , Bm^′ ) is called BSVN spanning subgraph-structure of BSVNGS ˇG^bn= (B, B1, B2, . . . , B^m) if B^′ = B and

Tk^′P(b, d) ≤ Tk^P(b, d), Ik^′P(b, d) ≤ Ik^P(b, d), Fk^′P(b, d) ≥ Fk^P(b, d),

Tk^′N(b, d) ≥ Tk^N(b, d), Ik^′N(b, d) ≥ Ik^N(b, d), Fk^′N(b, d) ≤ Fk^N(b, d), k = 1, 2, . . . , m.

Example 2.12. A BSVNGS represented in Fig. 4 is a BSVN spanning subgraph- structure of BSVNGS represented in Fig. 1.

Definition 2.13. Let ˇG^bn= (B, B1, B2, . . . , B^m) be a BSVNGS. Then bd ∈ B^k is called a BSVN B^k-edge or shortly B^k-edge, if

Tk^P(b, d) > 0 or Ik^P(b, d) > 0 or Fk^P(b, d) > 0 or Tk^N(b, d) < 0 or Ik^N(b, d) < 0 or Fk^N(b, d) < 0 or all these conditions are satisfied.

Consequently support of Bk is defined as:

b b b

b

B

1 ′

(0.1 ,0.2

,0.4 ,−0.

1,− 0.2,−

0.4) b₄

(0.3, 0.2

, 0.3, − 0.3, −

0.2, − 0.3)

b³(0.3,0.4, 0.3,−0.3,−0.4,−0.3) b²(0.2,0.2,0.3,−

0.2,−0.2,−0.3)

b₁

(0.2, 0.3, 0.4

, −0.2, −0.3, −0.4)

B^′2(0.2, 0.1, 0.4,−0.2,−0.1,−0.4) B_′

1(0 .2,0.1, 0

.3,− 0.2, −

0.1, − 0.3) B^′

2(0.1,0.1,0.3, −

0.1,−

0.1,−

0.3) B^′1(0.1,0.1, 0.4,−0.1,−0.1,−0.4)

Figure 4. A BSVN spanning subgraph-structure

supp(B^k) = {bd ∈ B^k : Tk^P(b, d) > 0} ∪ {bd ∈ B^k : Ik^P(b, d) > 0} ∪ {bd ∈ B^k : Fk^P(b, d) > 0} ∪ {bd ∈ B^k : Tk^N(b, d) < 0} ∪ {bd ∈ B^k : Ik^N(b, d) < 0} ∪ {bd ∈ B^k : Fk^N(b, d) < 0}, k = 1, 2, . . . , m.

Definition 2.14. B^k-path in BSVNGS ˇG^bn = (B, B1, B2, . . . , B^m) is a sequence b1, b2, . . . , b^mof distinct nodes(vertices) (except b^m= b1) in V such that b^k−1b^k is a BSVN B^k-edge for all k = 2, . . . , m.

Definition 2.15. A BSVNGS ˇG^bn = (B, B1, B2, . . . , B^m) is B^k-strong for any k ∈ {1, 2, . . . , m} if

Tk^P(b, d) = min{T^P(b), T^P(d)}, Ik^P(b, d) = min{I^P(b), I^P(d)}, Fk^P(b, d) = max{F^P(b), F^P(d)}, Tk^N(b, d) = max{T^N(b), T^N(d)},

Ik^N(b, d) = max{I^N(b), I^N(d)}, Fk^N(b, d) = min{F^N(b), F^N(d)},

∀ bd ∈ supp(Bk). If ˇGbn is Bk-strong for all k ∈ {1, 2, . . . , m}, then ˇGbn is called strong BSVNGS.

Example 2.16. Consider BSVNGS ˇG^bn = (B, B1, B2, B3) as depicted in Fig. 5.

Then ˇG^bnis strong BSVNGS, since it is B1−, B2− and B3− strong.

Definition 2.17. A BSVNGS ˇGbn= (B, B1, B2, . . . , Bm) is called complete BSVNGS if

b

b b b b

b

b b

b³(0.3, 0.3, 0.3,−0.3,−0.3,

−0.3)

b⁷(0.4, 0.4, 0.3,−0.4,−0.4,

−0.3)

b1(0.4, 0.3, 0.5, −0.4, −0.3, −0.5)

b²(0.3,0.2, 0.4,−0.3,−0.2,−0.4) b4(0.4,0.3,0.5,−0.4,−0.3,−0.5) b5(0.2,0.1,0.3,−0.2,−0.1,−0.3)

b⁶(0.3,0.3, 0.6,−0.3,−0.3,−0.6)

b8(0.2, 0.3, 0.4, −0.2, −0.3, −0.4)

B¹(0.2, 0.3,0.6, −0.2, −0.3, −0.6)

B²

(0.2, 0.3, 0.4,−0.2,−0.3,−0.4)

B1(0.3,0.2,0.4,−0.3,−0.2,−0.4)

B2(0.2,0.1,0.5,−0.2,−0.1,−0.5)

B¹(0.3, 0.2,0.5, −0.3, −0.2, −0.5)

B¹(0.3, 0.3,0.5, −0.3, −0.3, −0.5) B²(0.2, 0.1,0.3, −0.2, −0.1, −0.3)

B1(0.2,0.3,0.5,−0.2,−0.3,−0.5)

B¹

(0.2, 0.1, 0.6,−0.2,−0.1,−0.6)

B³ (0.3, 0.2, 0.5,−0.3,

−0.2,

−0.5)

B³ (0.3, 0.3, 0.5,−0.3,

−0.3,

−0.5) B2(0.3,0.3,0.6,−0.3,−0.3,−0.6)

Figure 5. A Strong BSVNGS

(1) ˇG^bnis strong BSVNGS.

(2) supp(B^k) 6= ∅, for all k = 1, 2, . . . , m.

(3) For all b, d ∈ V , bd is a B^k− edge for some k.

Example 2.18. Let ˇG^bn = (B, B1, B2) be BSVNGS of GS ˇG = (V, V1, V2), such that V = {b1, b2, b3, b4}, V1= {b1b2, b3b4}, V2= {b1b3, b2b3, b1b4, b2b4}.

Through direct calculations it is easily shown that ˇGbn is strong BSVNGS. More- over, supp(B1) 6= ∅, supp(B2) 6= ∅ and each pair b^kb^l of nodes in V is either a B1-edge or B2-edge. Hence ˇG^bnis complete BSVNGS, that is, B1− B2−complete BSVNGS.

Definition 2.19. Let ˇG^b1= (B1, B11, B12, . . . , B1m) and ˇG^b2= (B2, B21, B22, . . . , B2m) be two BSVNGSs. Lexicographic product of ˇG^b1 and ˇG^b2, denoted by

Gˇb1• ˇGb2 = (B1• B2, B11• B21, B12• B22, . . . , B1m• B2m), is defined as:

bb b

bB2(0.2, 0.2, 0.4, −0.2, −0.2, −0.4)

B² (0.1, 0.3, 0.5, −

0.1,− 0.4,−

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

B

1(0.2 , 0.3

, 0.5,

−0.2,

−0.3,

−0.5) B²(0.2,0.2,0.5,−

0.2,−0.2,−0.5)

B

1(0.1 ,0.2

,0.5 ,−0.1

,−0.2 ,−0.5)

B2(0.1,0.4,0.5,−0.1,−0.4,−0.5) b2(0.2,0.3,0.3,−0.2,−0.3,−0.3) b4(0.2,0.2,0.4,−0.2,−0.2,−0.4)

b3(0.1, 0.4, 0.5, −0.1, −0.4, −0.5)

Figure 6. A complete BSVNGS

(i)





 T_(B^P

1•B

2)(bd) = (TB^P₁• TB^P₂)(bd) = TB^P₁(b) ∧ TB^P₂(d) I_(B^P

1•B

2)(bd) = (IB^P₁ • IB^P₂)(bd) = IB^P₁(b) ∧ IB^P₂(d) F_(B^P

1•B

2)(bd) = (FB^P₁• FB^P₂)(bd) = FB^P₁(b) ∨ FB^P₂(d) (ii)





 T_(B^N

1•B₂)(bd) = (TB^N

1• TB^N

2)(bd) = TB^N

1(b) ∨ TB^N

2(d) I_(B^N

1•B₂)(bd) = (IB^N

1 • IB^N

2)(bd) = IB^N

1(b) ∨ IB^N

2(d) F_(B^N

1•B₂)(bd) = (FB^N

1• FB^N

2)(bd) = FB^N

1(b) ∧ FB^N

2(d) for all (bd) ∈ V1× V2,

(iii)





 T_(B^P

1k•B

2k)(bd1)(bd2) = (TB^P

1k• TB^P

2k)(bd1)(bd2) = TB^P

1(b) ∧ TB^P

2k(d1d2) I_(B^P

1k•B_2k)(bd1)(bd2) = (IB^P

1k• IB^P

2k)(bd1)(bd2) = IB^P

1(b) ∧ IB^P

2k(d1d2) F_(B^P

1k•B

2k)(bd1)(bd2) = (FB^P

1k• FB^P

2k)(bd1)(bd2) = FB^P

1(b) ∨ FB^P

2k(d1d2) (iv)





 T_(B^N

1k•B

2k)(bd1)(bd2) = (TB^N

1k• TB^N

2k)(bd1)(bd2) = TB^N

1(b) ∨ TB^N

2k(d1d2) I_(B^N

1k•B

2k)(bd1)(bd2) = (IB^N

1k• IB^N

2k)(bd1)(bd2) = IB^N

1(b) ∨ IB^N

2k(d1d2) F_(B^N

1k•B

2k)(bd1)(bd2) = (FB^N

1k• FB^N

2k)(bd1)(bd2) = FB^N

1(b) ∧ FB^N

2k(d1d2) for all b ∈ V1 , (d1d2) ∈ V2k,

(v)





 T_(B^P

1k•B

2k)(b1d1)(b2d2) = (TB^P

1k• TB^P

2k)(b1d1)(b2d2) = TB^P

1k(b1b2) ∧ TB^P

2k(d1d2) I_(B^P

1k•B

2k)(b1d1)(b2d2) = (IB^P

1k• IB^P

2k)(b1d1)(b2d2) = IB^P

1k(b1b2) ∧ IB^P

2k(d1d2) F_(B^P

1k•B

2k)(b1d1)(b2d2) = (FB^P

1k• FB^P

2k)(b1d1)(b2d2) = FB^P

1k(b1b2) ∨ FB^P

2k(d1d2)

b b

b b

B 12 (0.4

,0.2 ,0.6,−

0.4,− 0.2,−

0.6) B¹

1(0.5,0 .1,0.8,−0.5,−0.1,−0.8)

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

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

b

B 21 (0.2

,0.1 ,0.4,

− 0.2,−

0.1,− 0.4) B²

2(0.3,0 .2,0

.5,−0.3,−0.2,−0.5)

d2(0.3, 0.2, 0.4, −0.3, −0.2, −0.4) d3(0.5,0.3,0.5,−0.5,−0.3,−0.5) d1(0.2,0.1,0.3,−0.2,−0.1,−0.3)

Figure 7. Two BSVNGSs ˇG^b1and ˇG^b2

b

b¹d³(0.5,0.1, 0.6,−0.5,−0.1,−0.6)

b

b b b

b₁ d₁

(0.2, 0.1 , 0.6, −

0.2, − 0.1, −

0.6)

b₂d₂ (0.3,

0.2, 0.8, −

0.3,−0.2,−0.8) b₂

d₃ (0.5, 0.2

, 0.8, − 0.5, −

0.2, − 0.8) b¹d²(0.3,0.1, 0.6,−0.3,−0.1,−0.6)

b2d1(0.2, 0.1, 0.8, −0.2,−0.1, −0.8)

B¹²

•B²²

(0.3, 0.2, 0.8,−0 .3,−0

.2,−0 .8)

B12

• B22(0.3, 0.1,0.6,

−0.3, − 0.1,−0.6) B11•B21(0.2, 0.1, 0.6, −0.2, −0.1, −0.6)

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

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

b

(vi)





 T_(B^N

1k•B

2k)(b1d1)(b2d2) = (TB^N

1k• TB^N

2k)(b1d1)(b2d2) = TB^N

1k(b1b2) ∨ TB^N

2k(d1d2) I_(B^N

1k•B

2k)(b1d1)(b2d2) = (IB^N

1k• IB^N

2k)(b1d1)(b2d2) = IB^N

1k(b1b2) ∨ IB^N

2k(d1d2) F_(B^N

1k•B

2k)(b1d1)(b2d2) = (FB^N

1k• FB^N

2k)(b1d1)(b2d2) = FB^N

1k(b1b2) ∧ FB^N

2k(d1d2) for all (b1b2) ∈ V1k , (d1d2) ∈ V2k.

Example 2.20. Consider ˇG^b1 = (B1, B11, B12) and ˇG^b2 = (B2, B21, B22) are two BSVNGSs of GSs ˇG^s1 = (V1, V11, V12) and ˇG^s2 = (V2, V21, V22), respectively, as depicted in Fig. 7, where V11= {b1b2}, V12= {b3b4}, V21= {d1d2}, V22= {d2d3}.

Lexicographic product of BSVNGSs ˇG^b1and ˇG^b2 shown in Fig. 7 is defined as:

Gˇ^b1• ˇG^b2 = {B1• B2, B11• B21, B12• B22} and is depicted in Fig. 8.

b

b⁴d²(0.3, 0.2, 0.6, −0.3, −0.2, −0.6)

b

b b b

b₃ d₁

(0.2, 0.1 , 0.4, −

0.2,−0.1, − 0.4)

b3d3(0.4, 0.2, 0.5, −0.4,−0.2, −0.5)

b₄ d₃

(0.5, 0.2 , 0.6, −

0.5, − 0.2, −

0.6) b⁴d¹(0.2, 0.1, 0.6,−0.2,−0.1,−0

.6) b3d2

(0.3, 0

.2, 0.4, −0.3, −0.2, −0.4)

B11• B 21(0.2, 0.1,

0.6,−0.2, − 0.1,−0.6) B¹¹

•B²¹

(0.2, 0.1, 0.4,−0 .2,−0

.1,−0 .4)

B12•B22(0.3,0.2,0.5,−0.3,−0.2,−0.5) b

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

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

Figure 8. ˇG^b1• ˇG^b2

Theorem 2.21. Lexicographic product ˇG^b1• ˇG^b2 = (B1• B2, B11• B21, B12• B22, . . . , B1m• B2m) of two BSVNSGSs of GSs Gˇ^s1 and Gˇ^s2 is a BSVNGS of Gˇ^s1• ˇG^s2.

proof. Consider two cases:

Case 1.: For b ∈ V1, d1d2∈ V2k

T_(B^P

1k•B

2k)((bd1)(bd2)) = TB^P

1(b) ∧ TB^P

2k(d1d2)

≤ TB^P

1(b) ∧ [TB^P

2(d1) ∧ TB^P

2(d2)]

= [TB^P₁(b) ∧ TB^P₂(d1)] ∧ [TB^P₁(b) ∧ TB^P₂(d2)]

= T_(B^P₁•B₂)(bd1) ∧ T_(B^P₁•B₂)(bd2), T_(B^N

1k•B

2k)((bd1)(bd2)) = TB^N

1(b) ∨ TB^N

2k(d1d2)

≥ TB^N

1(b) ∨ [TB^N

2(d1) ∨ TB^N

2(d2)]

= [TB^N₁(b) ∨ TB^N₂(d1)] ∨ [TB^N₁(b) ∨ TB^N₂(d2)]

= T_(B^N₁•B

2)(bd1) ∨ T_(B^N₁•B

2)(bd2),

I_(B^P

1k•B

2k)((bd1)(bd2)) = IB^P₁(b) ∧ IB^P

2k(d1d2)

≤ IB^P₁(b) ∧ [IB^P₂(d1) ∧ IB^P₂(d2)]

= [IB^P

1(b) ∧ IB^P

2(d1)] ∧ [IB^P

1(b) ∧ IB^P

2(d2)]

= I_(B^P₁•B₂)(bd1) ∧ I_(B^P₁•B₂)(bd2),

I_(B^N_1k•B

2k)((bd1)(bd2)) = IB^N

1(b) ∨ IB^N

2k(d1d2)

≥ IB^N

1(b) ∨ [IB^N

2(d1) ∨ IB^N

2(d2)]

= [IB^N₁(b) ∨ IB^N₂(d1)] ∨ [IB^N₁(b) ∨ IB^N₂(d2)]

= I_(B^N

1•B

2)(bd1) ∨ I_(B^N

1•B

2)(bd2),

F_(B^P

1k•B

2k)((bd1)(bd2)) = FB^P₁(b) ∨ FB^P

2k(d1d2)

≤ FB^P₁(b) ∨ [FB^P₂(d1) ∨ FB^P₂(d2)]

= [FB^P

1(b) ∨ FB^P

2(d1)] ∨ [FB^P

1(b) ∨ FB^P

2(d2)]

= F_(B^P₁•B₂)(bd1) ∨ F_(B^P₁•B₂)(bd2),

F_(B^N

1k•B

2k)((bd1)(bd2)) = FB^N₁(b) ∧ FB^N_2k(d1d2)

≥ FB^N

1(b) ∧ [FB^N

2(d1) ∧ FB^N

2(d2)]

= [FB^N

1(b) ∧ FB^N

2(d1)] ∧ [FB^N

1(b) ∧ FB^N

2(d2)]

= F_(B^N

1•B

2)(bd1) ∧ F_(B^N

1•B

2)(bd2), for bd1, bd2∈ V1• V2.

Case 2.: For b1b2∈ V1k, d1d2∈ V2k

T_(B^P

1k•B

2k)((b1d1)(b2d2)) = TB^P

1k(b1b2) ∧ TB^P

2k(d1d2)

≤ [TB^P₁(b1) ∧ TB^P₁(b2] ∧ [TB^P₂(d1) ∧ TB^P₂(d2)]

= [TB^P

1(b1) ∧ TB^P

2(d1)] ∧ [TB^P

1(b2) ∧ TB^P

2(d2)]

= T_(B^P₁•B₂)(b1d1) ∧ T_(B^P₁•B₂)(b2d2),

T_(B^N

1k•B

2k)((b1d1)(b2d2)) = TB^N_1k(b1b2) ∨ TB^N_2k(d1d2)

≥ [TB^N

1(b1) ∨ TB^N

1(b2] ∨ [TB^N

2(d1) ∨ TB^N

2(d2)]

= [TB^N

1(b1) ∨ TB^N

2(d1)] ∨ [TB^N

1(b2) ∨ TB^N

2(d2)]

= T_(B^N

1•B

2)(b1d1) ∨ T_(B^N

1•B

2)(b2d2),

I_(B^P

1k•B

2k)((b1d1)(b2d2)) = IB^P

1k(b1b2) ∧ IB^P

2k(d1d2)

≤ [IB^P₁(b1) ∧ IB^P₁(b2] ∧ [IB^P₂(d1) ∧ IB^P₂(d2)]

= [IB^P

1(b1) ∧ IB^P

2(d1)] ∧ [IB^P

1(b2) ∧ IB^P

2(d2)]

= I_(B^P₁•B₂)(b1d1) ∧ I_(B^P₁•B₂)(b2d2),