Single-Valued Neutrosophic Graph Structures

Single-Valued Neutrosophic Graph Structures Muhammad Akram and Muzzamal Sitara Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan, E-mail: makrammath@yahoo.com,

m.akram@pucit.edu.pk, muzzamal.sitara@gmail.com

Abstract A graph structure is a generalization of undirected graph which is quite useful in studying some structures, including graphs and signed graphs. In this research paper, we apply the idea of single-valued neutrosophic sets to graph structure, and explore some interesting properties of single-valued neutrosophic graph structure.

We also discuss the concept of φ-complement of single-valued neutrosophic graph structure.

Key-words: Graph structure, Single-valued Neutrosophic graph structure, φ-complement.

Mathematics Subject Classification 2000: 03E72, 05C72, 05C78, 05C99

1

Introduction

Fuzzy set theory was introduced by Zadeh [14] to solve problems with uncertainties. At present, in modeling and controlling unsure systems in industry, society and nature, fuzzy sets and fuzzy logic are playing a vital role. In decision making, they can be used as power full mathematical tools which facilitate for approximate reasoning.

They play a significant role in complex phenomena which is not easily described by classical mathematics.

Atanassov [3] illustrated the extension of fuzzy sets by adding a new component, called, intuitionistic fuzzy sets. The intuitionistic fuzzy sets have essentially higher describing possibilities than fuzzy sets. The idea of intuitionistic fuzzy set is more meaningful as well as inventive due to the presence of degree of truth, degree of false and the hesitation margin. The hesitation margin of intuitionistic fuzzy set is its indeterminacy value by default. Samrandache [10] submitted the idea of neutrosophic set (NS)by combining the non-standard analysis, a tricomponent logic/set/probability theory and philosophy. “It is a branch of philosophy which studies the origin, nature and scope of neutralities as well as their interactions with different ideational spectra” [11]. A NS has three components: truth membership, indeterminacy membership and falsity membership, in which each membership value is a real standard or non-standard subset of the nonstandard unit interval ]0−, 1 + [ ([10]). To apply NSs in real-life problems more conveniently, Wang et al. [12] defined single-valued neutrosophic sets (SVNSs). A SVNS is a generalization of intuitionistic fuzzy sets [3]. In SVNS three components are not dependent and their values are contained in the standard unit interval [0, 1].

Fuzzy graphs were narrated by Rosenfeld [8] in 1975. Dinesh and Ramakrishnan [5] introduced the notion of a fuzzy graph structure and discussed some related properties. Akram and Akmal [1] introduced the concept of bipolar fuzzy graph structures. Broum et al. [4] portrayed single-valued neutrosophic graphs. Akram and Shahzadi [2] introduced the notion of neutrosophic soft graphs with applications. In this research paper, we apply the idea of single-valued neutrosophic sets to graph structure, and explore some interesting properties of single-valued neutrosophic graphs. We also discuss the concept of φ-complement of single-valued neutrosophic graph structure.

2

Single-Valued Neutrosophic Graph Structures

ˇ n = (Q, Q1 , Q2 , ..., Qn ) is called a single-valued neutrosophic graph structure (SVNSGS) of a Definition 2.1. G ˇ graph stucture G = (S, S1 , S2 ..., Sn ) if Q =< (m, n), T (m, n), I(m, n), F (m, n) > is a single-valued neutrosophic (SVNS) set on S and Qi =< n, Ti (n), Ii (n), Fi (n) > is a single-valued neutrosophic set on Si such that Ti (m, n) ≤ min{T (m), T (n)}, Ii (m, n) ≤ min{I(m), I(n)}, Fi (m, n) ≤ max{F (m), F (n)}, ∀m, n ∈ S. 1

Note that Ti (m, n) = 0 = Ii (m, n) = Fi (m, n) for all (m, n) ∈ S × S − Si and 0 ≤ Ti (m, n) + Ii (m, n) + Fi (m, n) ≤ 3 ∀(m, n) ∈ Si , where S and Si (i = 1,2,...,n ) are underlying vertex and ˇ n respectively. underlying i-edge sets of G ˇ n = (Q, Q1 , Q2 , ..., Qn ) be a SVNSGS of G. ˇ If H ˇ n = (Q′ , Q′ , Q′ , ..., Q′ ) is a SVNSGS Definition 2.2. Let G 1 2 n ˇ of G such that T ′ (n) ≤ T (n), I ′ (n) ≤ I(n), F ′ (n) ≥ F (n) ∀n ∈ S, Ti′ (m, n) ≤ Ti (m, n), Ii′ (m, n) ≤ Ii (m, n), Fi′ (m, n) ≥ Fi (m, n), ∀mn ∈ Si , i = 1, 2, ..., n. ˇ n is called a SVNS subgraph structure of SVNSGS G ˇn.

Then H ˇ n = (Q′ , Q′1 , Q′2 , ..., Q′n ) is called a SVNS induced subgraph structure of G ˇ n by Definition 2.3. A SVNSGS H a subset R of S if T ′ (n) = T (n), I ′ (n) = I(n), F ′ (n) = F (n) ∀n ∈ R, Ti′ (m, n) = Ti (m, n), Ii′ (m, n) = Ii (m, n), Fi′ (m, n) = Fi (m, n), ∀m, n ∈ R, i = 1, 2, ..., n. ˇ n = (Q′ , Q′1 , Q′2 , ..., Q′n ) is called a SVNS spanning subgraph structure of G ˇ n if Definition 2.4. A SVNSGS H ′ Q = Q and Ti′ (m, n) ≤ Ti (m, n), Ii′ (m, n) ≤ Ii (m, n), Fi′ (m, n) ≥ Fi (m, n), i = 1, 2, ..., n. ˇ = (S, S1 , S2 ) and Q, Q1 ,Q2 be SVNS subsets of S, S1 , S2 respectively, such that Example 2.5. Consider a GS G Q = {(n1 , .5, .2, .3), (n2 , .7, .3, .4), (n3 , .4, .3, .5), (n4 , .7, .3, .6)}, Q1 = {(n1 n2 , .5, .2, .4), (n2 n4 , .7, .3, .6)}, Q2 = {(n3 n4 , .4, .3, .6), (n1 n4 , .5, .2, .6)}. ˇ n = (Q, Q1 , Q2 ) is a SVNSGS of G ˇ as shown in Fig. 2.1.

Direct calculations show that G n1 (.5, .2, .3)

n2 (.7, .3, .4)

b

b

Q2 (.5, .2, .6)

Q1 (.5, .2, .4)

Q b

(.7

, .3

, .6

)

1

Q2 (.4, .3, .6) b

n4 (.7, .3, .6)

n3 (.4, .3, .5)

Figure 2.1: A single-valued neutrosophic graph structure ˇ n = (Q′ , Q11 , Q12 ) is a SVNS subgraph structure of G ˇ n as shown in Fig. 2.2.

Example 2.6. SVNSGS K n1 (.4, .1, .4)

n2 (.6, .2, .5)

b

b

Q12 (.4, .1, .7)

Q11 (.4, .1, .5)

(.5

Q b

, .2

, .7

)

11

Q12 (.3, .2, .8)

n4 (.6, .2, .7)

b

n3 (.3, .2, .6)

ˇn Figure 2.2: A SVNS subgraph structure K ˇ n = (Q, Q1 , Q2 , ..., Qn ) be a SVNSGS of G. ˇ Then mn ∈ Si is called a SVNS Qi -edge Definition 2.7. Let G or simply Qi -edge, if Ti (m, n) > 0 or Ii (m, n) > 0 or Fi (m, n) > 0 or all three conditions hold. Consequently, support of Qi ; i=1,2,...,n is: supp(Qi ) = {mn ∈ Qi : Ti (m, n) > 0} ∪ {mn ∈ Qi : Ii (m, n) > 0} ∪ {mn ∈ Qi : Fi (m, n) > 0}. ˇ n = (Q, Q1 , Q2 , ..., Qn ) is a sequence of distinct vertices n1 , n2 , ..., nm Definition 2.8. Qi -path in a SVNSGS G (except choice that nm = n1 ) in S, such that nj−i nj is a SVNS Qi -edge ∀j = 1, 2, ..., m.

2

ˇ n = (Q, Q1 , Q2 , ..., Qn ) is called Qi -strong for some i ∈ {1, 2, ..., n} if Definition 2.9. A SVNSGS G Ti (m, n) = min{T (m), T (n)}, Ii (m, n) = min{I(m), I(n)}, Fi (m, n) = max{F (m), F (n)}, ∀mn ∈ supp(Qi ). ˇ n is called strong if it is Qi -strong for all i ∈ {1, 2, ..., n}.

SVNSGS G ˇ n = (Q, Q1 , Q2 ) as shown in Fig. 2.3. Then G ˇ n is a strong SVNSGS Example 2.10. Consider SVNSGS G since it is both Q1 − and Q2 − strong.

, (.2

n3 (.2, .6, .5) b

b

n4 (.4, .4, .5)

b

)

Q1 (.2, .6, .5)

Q1 (.1, .6, .7) b

n2 (.3, .7, .6)

.6 .6 ,

Q2

Q1 (.2, .4, .5)

Q2 (.3, .4, .5) n5 (.3, .5, .4)

Q2 (.3, .6, .6)

n1 (.3, .6, .5) b .5 )

.4 ,

Q2 (.2, .4, .6)

Q

.3 , 1(

n8 (.1, .8, .7) ) b , .5, .7 Q 2(.1 Q2 (.3, .5, .5)

Q1 (.1, .4, .7)

.2, .4 Q 1(

b

n6 (.6, .6, .5)

n7 (.2, .4, .3) b

, .5)

ˇ n = (Q, Q1 , Q2 ) Figure 2.3: A strong SVNSGS G ˇ n = (Q, Q1 , Q2 , ..., Qn ) is called complete or Q1 Q2 ...Qn -complete, if G ˇ n is a Definition 2.11. A SVNSGS G strong SVNSGS, supp(Qi ) 6= φ for all i=1,2,...,n and for every pair of vertices m, n ∈ S, mn is an Qi − edge for some i. ˇ n = (Q, Q1 , Q2 ) be a SVNSGS of graph structure G ˇ = (S, S1 , S2 ) such that S = Example 2.12. Let G {n1 , n2 , n3 }, S1 = {n1 n2 }, S2 = {n2 n3 , n1 n3 } as shown in Fig. 2.4. by simple calculations, it can be seen that ˇ n is a strong SVNSGS. G n1 (.5, .4, .5) b

Q

.8 )

2 (.

.4 ,

5, .

4, .

6)

, (.4 Q1

Q2 (.4, .7, .8) b

b

n2 (.4, .7, .8)

n3 (.5, .7, .6)

Figure 2.4: A complte SVNSGS Moreover, supp(Q1 ) 6= φ, supp(Q2 ) 6= φ and each pair of vertices in S, is either a Q1 -edge or an Q2 -edge. ˇ n is a complete, i.e., Q1 Q2 -complete SVNSGS. So G ˇ n = (Q, Q1 , Q2 , ..., Qn ) be a SVNSGS. Then truth strength, indeterminacy strength Definition 2.13. Let G and falsity strength of a Qi -path PQi =n1 , n2 , ..., nm are denoted by T.PQi , I.PQi and F.PQi respectively and defined as m m m W V V P [FQPi (nj−1 nj )].

[IQ (nj−1 nj )], F.PQi = [TQPi (nj−1 nj )], I.PQi = T.PQi = i j=2

j=2

j=2

ˇ n = (Q, Q1 , Q2 ) as shown in Fig. 2.4. We found that Example 2.14. Consider a SVNSGS G PQ2 = n2 , n1 , n3 is a Q2 -path. So T.PQ2 = 0.4, I.PQ2 = 0.4 and F.PQ2 = 0.8. ˇ n = (Q, Q1 , Q2 , ..., Qn ) be a SVNSGS. Then Definition 2.15. Let G

W {TQj i (mn)}, such that TQj i (mn) Qi -truth strength of connectedness between m and n is defined by TQ∞i (mn) = j≥1 W ◦ TQ1 i )(mn) for j ≥ 2 and TQ2 i (mn) = (TQ1 i ◦ TQ1 i )(mn) = (TQ1 i (mz) ∧ TQ1 i )(zn). = (TQj−1 i z

W j ∞ {IQi (mn)}, such that Qi -indeterminacy strength of connectedness between m and n is defined by IQ (mn) = i j≥1 W 1 j j−1 1 1 2 1 1 IQ (mz) ∧ IQ (mn) = (IQ )(zn). ◦ IQ )(mn) for j ≥ 2 and IQ (mn) = (IQ ◦ IQ )(mn) = (IQ i i i i i i i i z

3

V {FQj i (mn)}, such that Qi -Falsity strength of connectedness between m and n is defined by FQ∞i (mn) = j≥1 V 1 1 2 1 1 (F FQj i (mn) = (FQj−1 (mz) ∨ FQ1 i )(zn). ◦ F )(mn) for j ≥ 2 and F (mn) = (F ◦ F )(mn) = Qi Qi Qi Qi Qi i z

ˇ n = (Q, Q1 , Q2 , ..., Qn ) is a Qi -cycle if (supp(Q), supp(Q1), supp(Q2 ), ..., supp(Qn )) Definition 2.16. A SVNSGS G is a Qi − cycle. ˇ n = (Q, Q1 , Q2 , ..., Qn ) is a SVNS fuzzy Qi -cycle (for some i) if G ˇ n is a Qi -cycle, Definition 2.17. A SVNSGS G ˇ n such that no unique Qi -edge mn is in G TQi (mn) = min{TQi (rs) : rs ∈ Si = supp(Qi )} or IQi (mn) = min{IQi (rs) : rs ∈ Si = supp(Qi )} or FQi (mn) = max{FQi (rs) : rs ∈ Si = supp(Qi )}. ˇ n = (Q, Q1 , Q2 ) as shown in Fig. 2.3. Then G ˇ n is a Q1 -cycle and SVNS Example 2.18. Consider a SVNSGS G fuzzy Q1 − cycle, since (supp(Q), supp(Q1 ), supp(Q2 )) is a Q1 -cycle and there is no unique Q1 -edge satisfying above condition. ˇ n = (Q, Q1 , Q2 , ..., Qn ) be a SVNSGS and p be a vertex in G ˇ n . Let (Q′ , Q′ , Q′ , ..., Q′ ) Definition 2.19. Let G 1 2 n be a SVNSGS induced by S \ {p} such that ∀v 6= p, w 6= p ˇ n . TQ′ (v) = TQ (v), TQ′ (p) = 0 = IQ′ (p) = FQ′ (p), TQ′i (pv) = 0 = IQ′i (pv) = FQ′i (pv) ∀ edges pv ∈ G IQ′ (v) = IQ (v), FQ′ (v) = FQ (v), ∀v 6= p. TQ′i (vw) = TQi (vw), IQ′i (vw) = IQi (vw), FQ′i (vw) = FQi (vw), Then p is SVNS fuzzy Qi -cut vertex for any i, if ∞ ∞ ∞ ∞ TQ∞i (vw) > TQ∞′ (vw), IQ (vw) > IQ ′ (vw) and FQ (vw) > FQ′ (vw), for some v, w ∈ S \ {p}. i i i i i ∞ Note that p is a Qi − T SVNS fuzzy cut vertex if TQi (vw) > TQ∞′ (vw), Qi − I SVNS fuzzy cut vertex if i ∞ ∞ ∞ ∞ IQ (vw) > IQ ′ (vw) and Qi − F SVNS fuzzy cut vertex if FQ (vw) > FQ′ (vw). i i i

i

ˇ n = (Q, Q1 , Q2 ) as shown in Fig.2.5 and G ˇ ′n = (Q′ , Q′1 , Q′2 ) be SVNS Example 2.20. Consider the SVNSGS G ˇ n found by deleting vertex n2 . Deleted vertex n2 is a SVNS fuzzy Q1 -I cut subgraph structure of SVNSGS G ∞ ∞ ∞ ∞ vertex since IQ (n n ) = 0.4 > 0.3 = IQ ′ (n2 n5 ), IQ (n3 n4 ) = 0.7 = IQ′ (n3 n4 ), 2 5 1 1 1 1 ∞ ∞ IQ (n3 n5 ) = 0.4 > 0.3 = IQ ′ (n3 n5 ).

1 1

ˇ n = (Q, Q1 , Q2 , ..., Qn ) be a SVNSGS and mn be Qi − edge.

Definition 2.21. Suppose G ˇ n , such that ∀ edges mn 6= rs, Let (Q′ , Q′1 , Q′2 , ..., Q′n ) be a SVNS fuzzy spanning subgraph structure of G TQ′i (mn) = 0 = IQ′i (mn) = FQ′i (mn), TQ′i (rs) = TQi (rs), IQ′i (rs) = IQi (rs), FQ′i (rs) = FQi (rs).

Then mn is a SVNS fuzzy Qi -bridge if ∞ ∞ ∞ ∞ (vw) > IQ TQ∞i (vw) > TQ∞′ (vw), IQ ′ (vw) and FQ (vw) > FQ′ (vw), for some v, w ∈ S. i i i i i ∞ (vw) > Note that mn is a Qi − T SVNS fuzzy bridge if TQ∞i (vw) > TQ∞′ (vw), Qi − I SVNS fuzzy bridge if IQ i i ∞ ∞ ∞ IQ′ (vw) and Qi − F SVNS fuzzy bridge if FQi (vw) > FQ′ (vw). i

i

ˇ n = (Q, Q1 , Q2 ) as shown in Fig. 2.5 and G ˇ ′ = (Q′ , Q′ , Q′ ) be SVNS Example 2.22. Consider the SVNSGS G 1 2 n ˇ spanning subgraph structure of SVNSGS Gn found by deleting Q1 -edge (n2 n5 ).

b

Q1 (0.3, 0.2, 0.4)

n4

Q1 (0.5, 0.7, 0.6)

n

3(

b

n1 Q2 ( (0.

0.3, 3, 0 0.6, 0.4) .6, 0.4 ) b

Q1 (0.4, 0.4, 0.5)

0.

5, 0.

7, 0.

6)

n2 (0.4, 0.7, 0.5)

.4) .5, 0 .3, 0 Q 1(0 Q2 (0.2, 0.4, 0.3)

Q2 (0.1, 0.4, 0.2) b b (0 4) .6 Q1 ( , 0. ,0 .2) 0 0.3, 0.4 , .9 .4 0.3, ,0 .3, ,0 .1 0 0 ( ( 0 b .4) .7 Q2 n6 ) n5 (0.4, 0.5, 0.6)

ˇ n = (Q, Q1 , Q2 ) Figure 2.5: A SVNSGS G Edge (n2 n5 ) is a SVNS fuzzy Q1 -bridge. Since TQ∞1 (n2 n5 ) = 0.4 > 0.3 = TQ∞′ (n2 n5 ), 1 ∞ ∞ ∞ = 0.4 > 0.3 = IQ ′ (n2 n5 ) and FQ (n2 n5 ) = 0.5 > 0 = FQ′ (n2 n5 ).

1

∞ IQ (n2 n5 ) 1

1

1

4

ˇ n = (Q, Q1 , Q2 , ..., Qn ) is a Qi -tree if (supp(Q), supp(Q1), supp(Q2 ), ..., supp(Qn )) Definition 2.23. A SVNSGS G ˇ n is a Qi -tree if a subgraph of G ˇ n induced by supp(Qi ) generates a tree. is a Qi − tree. In other words, G ˇ n = (Q, Q1 , Q2 , ..., Qn ) is a SVNS fuzzy Qi -tree if G ˇ n has a SVNS fuzzy Definition 2.24. A SVNSGS G ′′ ′′ ′′ ′′ ˇ ˇ n, spanning subgraph structure Hn = (Q , Q1 , Q2 , ..., Qn ) such that ∀Qi -edges mn not in H ∞ ∞ ∞ ′′ ˇ Hn is a Qi -tree, TQi (mn) < TQ′′ (mn), IQi (mn) < IQ′′ (mn), FQi (mn) < FQ′′ (mn). i i i ˇ n is a SVNS fuzzy Qi -T tree if TQi (mn) < T ∞′′ (mn), a SVNS fuzzy Qi -I tree if Inparticular, G Qi

∞ ∞ IQi (mn) < IQ ′′ (mn) and a SVNS fuzzy Qi -F tree if FQi (mn) > FQ′′ (mn). i

i

ˇ n = (Q, Q1 , Q2 ) as shown in Fig.2.6, which is a Q2 -tree. It is not a Example 2.25. Consider the SVNSGS G Q1 -tree but a SVNS fuzzy Q1 -tree since it has a single-vlaued neutrosophic fuzzy spanning subgraph (Q′ , Q′1 , Q′2 ) ˇ n . Moreover, as a Q′1 -tree, which is obtained by deleting Q1 -edge n2 n5 from G ∞ TQ1 (n2 n5 ) = 0.2 < 0.3 = TQ∞′ (n2 n5 ), IQ1 (n2 n5 ) = 0.1 < 0.3 = IQ ′ (n2 n5 and 1 1 ∞ FQ1 (n2 n5 ) = 0.6 > 0.5 = FQ1 ′ (n2 n5 ). n1 (.3, .6, .5) b

(.2,

Q2

n3 (.5, .7, .6)

Q2

4) .4 , .

Q2 (.3, .2, .3) b

(.2 ,

n6 (.3, .4, .3)

.6 ,

.2 )

(.

Q2

Q1 (.5, .7, .8)

n2 (.4, .7, .6) b

b

4, 1, .

.1 )

Q1 (.3, .5, .5)

Q

2 (.

1, .

4, .

Q1 (.2, .1, .6)

4)

b

b

n4 (.6, .9, .8)

n5 (.4, .5, .4

Q1 (.3, .3, .7)

Figure 2.6: A single-valued neutrosophic fuzzy Q1 -tree ˇ s1 = (Q1 , Q11 , Q12 , ..., Q1n ) of graph structure G ˇ 1 = (S1 , S11 , S12 , ..., S1n ) is Definition 2.26. A SVNSGS G ˇ ˇ isomorphic to SVNSGS Gs2 = (Q2 , Q21 , Q22 , ..., Q2n ) of graph structure G2 = (S2 , S21 , Q22 , ..., S2n ) if we have (f, φ) where f : S1 → S2 is a bijection and φ is a permutation on set {1, 2, ..., n} and following relations are satisfied; TQ1 (m) = TQ2 (f (m)), IQ1 (m) = IQ2 (f (m)), FQ1 (m) = FQ2 (f (m)), ∀m ∈ S1 and TQ1i (mn) = TQ2φ(i) (f (m)f (n)), IQ1i (mn) = IQ2φ(i) (f (m)f (n) FQ1i (mn) = FQ2φ(i) (f (m)f (n)), ∀mn ∈ S1i , i=1,2,...,n. ˇ n1 = (Q, Q1 , Q2 ) and G ˇ n2 = (Q′ , Q′ , Q′ ) be two SVNSGSs as shown in Fig. 2.7.

Example 2.27. Let G 1 2 m2 (.5, .5, .6)

n3 (.2, .7, .8)

b

) , .6 , .3 1 (.1

Q′

b b

.7 )

Q2

.6 )

.2 ,

(.1 ,

.3 ,

′ (.2, Q1

.3 , b

m1 (.3, .3, .4)

3 , .5

(.2, Q1

.6 )

m4 (.2, .3, .5)

)

n4 (.2, .3, .5)

Q ′( 2 .1 ,.

.3 , (.1 ,

b

.6 )

b

.3 ,

.7 )

Q1

.2 ,

n1 (.3, .3, .4)

b

, ′ (.2 Q2

, (.2 Q2

.5 )

b

b

n2 (.5, .5, .6)

m3 (.2, .7, .8)

Figure 2.7: Isomorphic SVNS graph structures ˇ n1 is isomorphic G ˇ n2 under (f, φ) where f : S → S ′ is a bijection and φ is a permutation on set {1, 2} G defined as φ(1) = 2, φ(2) = 1 and following relations are satisfied; TQ (ni ) = TQ′ (f (ni )), IQ (ni ) = IQ′ (f (ni )), FQ (ni ) = FQ′ (f (ni )), ∀ni ∈ S and TQi (ni nj ) = TQ′φ(i) (f (ni )f (nj )), IQi (ni nj ) = IQ′φ(i) (f (ni )f (nj )), FQi (ni nj ) = FQ′φ(i) (f (ni )f (nj )), ∀ni nj ∈ Si , i=1,2.

5

ˇ s1 = (Q1 , Q11 , Q12 , ..., Q1n ) of graph structure G ˇ 1 = (S1 , S11 , S12 , ..., S1n ) is Definition 2.28. A SVNSGS G ˇ ˇ identical to SVNSGS Gs2 = (Q2 , Q21 , Q22 , ..., Q2n ) of graph structure G2 = (S2 , S21 , Q22 , ..., S2n ) if f : S1 → S2 is a bijection and following relations are satisfied; TQ1 (m) = TQ2 (f (m)), IQ1 (m) = IQ2 (f (m)), FQ1 (m) = FQ2 (f (m)), ∀m ∈ S1 and TQ1i (mn) = TQ2i (f (m)f (n)), IQ1i (mn) = IQ2i (f (m)f (n)), FQ1i (mn) = FQ2(i) (f (m)f (n)), ∀mn ∈ S1i , i=1,2,...,n. ˇ n1 = (Q, Q1 , Q2 ) and G ˇ n2 = (Q′ , Q′ , Q′ ) be two SVNSGSs of GSs Example 2.29. Let Let G 1 2 ′ ′ ′ ˇ ˇ G1 = (S, S1 , S2 ), G2 = (S , S1 , S2 ) respectively as shown in Fig. 2.8 and Fig. 2.9. n5 (0.7, 0.6, 0.5) b Q2 (0 5) .6, 0.5 ,

, 0.4, 0.

Q 2(0.5

n3 (0.5, 0.4, 0.3)

4, (0.

Q1 n6 (0.4, 0.5, 0.2)

b

Q1 (0 .2, 0.2 ,

3) , 0.

0.4

0.4)

, 0.3) .1, 0.2 ) Q 2(0 0.4 .5, ,0 0.4 ( Q1 b

.4) , 0.3, 0 Q 1(0.2 Q1 (0 .3, 0.4 , 0.2)

b

5)

b

n8 (0.4, 0.6, 0.3)

.6) , 0.3, 0 Q 2(0.4 b

n4 (0.6, 0.5, 0.4)

n1 (0.2, 0.3, 0.4)

0.4, 0.5)

n2 (0.3,

, 0.3, 0.

Q 2(0.1 Q1 (0 .3, 0.2 , 0.5) b

0.5)

b

n7 (0.5, 0.3, 0.6)

ˇ n1 Figure 2.8: A SVNSGS G m5 (0.7, 0.6, 0.5) b

2 (0.

m1 (0.3, 0.4, 0.5)

2, 0 .

2, 0 .

4)

Q′

1 (0.

3, 0 .

m

b

1 (0.

, 0. ′ (0.1 Q2

.

2, 0

m2 (0.2, 0.3, 0.4)

2, 0 .

3(

5)

0.6 ,

Q′

.2,

1 (0.

b

1 (0

3) 0.

4, 0.

4,

Q′

m8 ( 0.4, 0.5, 0.2)

5, 0 .

0.4 )

, ′ (0.1 Q2

5) , 0.

0.3

6, 0 .

b

0.3 ,

b

0. ′ ( Q1

m4

Q′

Q′

5, (0.

3) , 0.

0.4

) , 0.5 , 0.4 ′ (0.5 Q2

3, 0 .

3)

0.5 ,0 .4) Q′ 1( 0.

4, 0 4 .2) ,0 .5 ,0 .4 ) b b

, 0.4

) , 0.6 , 0.3 ′ (0.4 Q2

5)

, 0.6

) 0.3

( m6

b

m7 (0.5, 0.3, 0.6)

ˇ n2 Figure 2.9: A SVNSGS G ˇ n1 is identical with G ˇ n2 under f : S → S ′ defined as ; SVNGS G f (n1 ) = m2 , f (n2 ) = m1 , f (n3 ) = m4 , f (n4 ) = m3 , f (n5 ) = m5 , f (n6 ) = m8 , f (n7 ) = m7 , f (n8 ) = m6 , TQ (ni ) = TQ′ (f (ni )), IQ (ni ) = IQ′ (f (ni )), FQ (ni ) = FQ′ (f (ni )), ∀ni ∈ S and TQi (ni nj ) = TQ′i (f (ni )f (nj )), IQi (ni nj ) = IQ′i (f (ni )f (nj )), FQi (ni nj ) = FQ′i (f (ni )f (nj )), ∀ni nj ∈ Si , i=1,2. ˇ n = (Q, Q1 , Q2 , ..., Qn ) be a SVNSGS and φ be a permutation on {Q1 , Q2 , ..., Qn } Definition 2.30. Suppose G and on {1, 2, ..., n} that is φ(QiW ) = Qj iff φ(i) = j ∀i. If mn ∈ Qi for any i and W TQφ (mn) = TQ (m) ∧ TQ (n) − Tφ(Qj ) (mn), IQφ (mn) = IQ (m) ∧ IQ (n) − Iφ(Qj ) (mn), i i j6=i j6=i V FQφ (mn) = FQ (m) ∨ FQ (n) − Tφ(Qj ) (mn), i = 1, 2, ..., n, then mn ∈ Qφk , where k is selected such that; i

j6=i

TQφ (mn) ≥ TQφ (mn), IQφ (mn) ≥ IQφ (mn), FQφ (mn) ≥ FQφ (mn) ∀i. k

i

i

k

i

k

ˇ n and symbolized as G ˇ φc And SVNSGS (Q, Qφ1 , Qφ2 , ..., Qφn ) is named as φ-complement of SVNSGS G n . ˇ Example 2.31. Let Gn = (Q, Q1 , Q2 , Q3 ) be a SVNSGS shown in Fig. 2.10, φ(1) = 2, φ(2) = 3, φ(3) = 1. As ˇ φc =(Q, Qφ , Qφ , Qφ ) is a result of simple calculations, n1 n3 ∈ Qφ3 , n2 n3 ∈ Qφ1 , n1 n2 ∈ Qφ2 So, G n 1 2 3 ˇ n as showm in Fig. 2.10. φ-complement of SVNSGS G 0.4) 0.3, 0.3, Q 3(

b

) , 0.7 , 0.4 φ (0.3 Q2 b

b

n2 (0.5, 0.6, 0.4)

n1 (0.3, 0.4, 0.7)

n1 (0.3, 0.4, 0.7) b Q1 ( 0.3, 0.4, 0.3)

Q2 (0.5, 0.4, 0.3)

Qφ 3 (0.

3, 0 .4, 0 .7) b

b

n3 (0.7, 0.5, 0.3)

n2 (0.5, 0.6, 0.4)

ˇn, G ˇ φc Figure 2.10: SVNSGSs G n 6

Qφ 1 (0.5, 0.5, 0.4)

n3 (0.7, 0.5, 0.3)

ˇ n = (Q, Q1 , Q2 , ..., Qn ) is always a strong SVNSGS. Proposition 2.32. A φ-complement of a SVNSGS G Moreover, if φ(i) = k, where i, k ∈ {1, 2, ..., n}, then all Qk -edges in SVNSGS (Q, Q1 , Q2 , ..., Qn ) become Qφi edges in (Q, Qφ1 , Qφ2 , ..., Qφn ).

Proof. According to the definition W W of φ-complement, Iφ(Qj ) (mn), Tφ(Qj ) (mn), IQφ (mn) = IQ (m) ∧ IQ (n) − TQφ (mn) = TQ (m) ∧ TQ (n) − i i j6=i j6= Vi Fφ(Qj ) (mn), for i ∈ {1, 2, ..., n}.

FQφ (mn) = FQ (m) ∨ FQ (n) − i

j6=i

For expression of truthness W in φ-complement requirements are shown as: Tφ(Qj ) (mn) ≥ 0 and TQi (mn) ≤ TQ (m) ∧ TQ (n) ∀Qi .

Since TQ (m) ∧ TQ (n) ≥ 0, j6=i W W Tφ(Qj ) (mn) ≥ 0.

Tφ(Qj ) (mn) ≤ TQ (m) ∧ TQ (n) ⇒ TQ (m) ∧ TQ (n) − ⇒

j6=i j6=i W Therefore, TQφ (mn) ≥ 0 ∀i. Moreover, TQφ (mn) achieves its maximum value when Tφ(Qj ) (mn) is zero. It i i j6=i W Tφ(Qj ) (mn) gets zero value. So, is obvious that when φ(Qi ) = Qk and mn is a Qk -edge then j6=i

TQφ (mn) = TQ (m) ∧ TQ (n), f or (mn) ∈ Qk , φ(Qi ) = Qk . Similarly, i IQφ (mn) = IQ (m) ∧ IQ (n), f or (mn) ∈ Qk , φ(Qi ) = Qk . i In the similar way for expression of falsity in φ-complement requirements are shown as: V Fφ(Qj ) (mn) ≥ 0 and FQi (mn) ≤ FQ (m) ∨ FQ (n) ∀Qi .

Since FQ (m) ∨ FQ (n) ≥ 0, j6=i V V Fφ(Qj ) (mn) ≥ 0.

Fφ(Qj ) (mn) ≤ FQ (m) ∨ FQ (n) ⇒ FQ (m) ∨ FQ (n) − ⇒

j6=i j6=i V Therefore, FQφ (mn) is non-negative for all i. Moreover, FQφ (mn) attains its maximum value when Fφ(Qj ) (mn) i i j6=i V becomes zero. It is clear that when φ(Qi ) = Qk and mn is a Qk -edge then Fφ(Qj ) (mn) gets zero value. So, j6=i

FQφ (mn) = FQ (m) ∨ FQ (n), f or (mn) ∈ Qk , φ(Qi ) = Qk . From these expressions of truthness, indeteminacy i and falsity required results are achieved. ˇ n = (Q, Q1 , Q2 , ..., Qn ) be a SVNSGS and φ a permutation on {1, 2, ..., n}. Then Definition 2.33. Let G ˇ n is isomorphic to G ˇ φc , then G ˇ n is said to be self-complementary. (i) If G n ˇ n is identical to G ˇ φc , then G ˇ n is said to be strong self-complementary. (ii) If G n ˇ n = (Q, Q1 , Q2 , ..., Qn ) be a SVNSGS. Then Definition 2.34. Suppose G ˇ n is isomorphic to G ˇ φc , ∀ permutations φ on {1,2,...,n}, then G ˇ n is totally self-complementary. (i) If G n ˇ n is totally self-complementary. ˇ n is identical to G ˇ φc , ∀ permutations φ on {1,2,...,n}, then G (ii) If G n Example 2.35. All strong SVNSGSs are self-complementary or totally self-complementary SVNSGSs. ˇ n = (Q, Q1 , Q2 , Q3 ) in Fig.2.11 is totally strong self-complementary SVNSGS. Example 2.36. SVNSGS G n2 (0.4, 0.5, 0.6)

Q1 (0.4, 0.4, 0.6)

b

n1 (0.7, 0.4, 0.5)

.4 , 3 (0

,0

.3 ,

b

0 .5

)

)

Q

0 .2

0 .5

n6 (0.4, 0.5, 0.6)

1(

.4 ,

b

Q

,0

0 .4

,0

.5 )

Q2 (0.2, 0.3, 0.5)

,0

4 (0.

Q2

Q

.2 (0 3

.3 ,0

.6 )

b

n4 (0.4, 0.5, 0.5)

b

b

n7 (0.2, 0.3, 0.4)

n5 (0.2, 0.3, 0.3)

b

n3 (0.2, 0.3, 0.4)

Figure 2.11: A totally strong self-complementary SVNSGS

7

Theorem 2.37. A SVNSGS is totally self-complementary if and only if it is strong SVNSGS. ˇ n and a permutation φ on {1,2,..., n}. By proposition 2.32, Proof. Consider a strong SVNSGS G ˇ φ-complement of a SVNSGS Gn = (Q, Q1 , Q2 , ..., Qn ) is always a strong SVNSGS. Moreover, if φ(i) = k, where i, k ∈ {1, 2, ..., n}, then all Qk -edges in SVNSGS (Q, Q1 , Q2 , ..., Qn ) become Qφi edges in (Q, Qφ1 , Qφ2 , ..., Qφn ).

This leads TQk (mn) = TQ (m) ∧ TQ (n) = TQφ (mn), IQk (mn) = IQ (m) ∧ IQ (n) = IQφ (mn), FQk (mn) = i i ˇ n and G ˇ φ are isomorphic FQ (m)∨FQ (n) = F φ (mn). Hence under the mapping(identity mapping) f : S → S, G n

Qi

such that,

TQ (m) = TQ (f (m)), IQ (m) = IQ (f (m)), FQ (m) = FQ (f (m)) and TQk (mn) = TQφ (f (m)f (n)) = TQφ (mn), i i IQk (mn) = IQφ (f (m)f (n)) = IQφ (mn) , FQk (mn) = FQφ (f (m)f (n)) = FQφ (mn) , i

i

i

i

ˇn ∀mn ∈ Sk , for φ (k) = i; i,k = 1,2,...,n. All this is satisfied for every permutation φ on {1,2,..., n}. Hence G φ ˇ ˇ is totally self-complementary SVNSGS. Conversely, let for every permutation φ on {1,2,..., n}, Gn and Gn are isomorphic. Then according to the definition of isomorphism of SVNSGSs and φ-complement of SVNSGS, −1

TQk (mn) = TQφ (f (m)f (n)) = TQ (f (m)) ∧ TQ (f (n)) = TQ (m) ∧ TQ (n), i

IQk (mn) = IQφ (f (m)f (n)) = IQ (f (m)) ∧ IQ (f (n)) = TQ (m) ∧ IQ (n), i

FQk (mn) = FQφ (f (m)f (n)) = FQ (f (m)) ∨ TQ (f (n)) = FQ (m) ∧ TQ (n), i

ˇ n is strong SVNSGS. ∀mn ∈ Sk , k = 1,2,...,n. Hence G Remark 2.38. Every SVNSGS which is self-complementary is definitely totally self-complementary. ˇ = (S, S1 , S2 , ..., Sn ) is a strong and totally self-complementary GS and Theorem 2.39. If G Q = (TQ , IQ , FQ ) is a SVNS subset of S where TQ , IQ , FQ are constant valued functions then a strong SVNSGS ˇ with SVNS vertex set Q is always a strong totally self-complementary SVNSGS. of G ˇ is Proof. Consider three constants p, q, r ∈ [0, 1], such that TQ (m) = p, IQ (m) = q, FQ (m) = r ∀m ∈ S Since G totally self-complementary strong GS, so there is a bijection f : S → S for any permutation φ−1 on {1,2,...,n}, ˇ ] is a Sk -edge in G ˇ φ−1 c . Hence for every Qk -edge such that for any Sk -edge (mn), (f(m)f(n)) [a Si -edge in G −1 ˇ n ] is a Qφ -edge in Gˇn φ c . (mn), (f(m)f(n)) [a Qi -edge in G k ˇ n is strong SVNSGS, so Moreover G TQ (m) = p = TQ (f (m)), IQ (m) = q = IQ (f (m)), FQ (m) = r = FQ (f (m)) ∀m ∈ S and TQk (mn) = TQ (m) ∧ TQ (n) = TQ (f (m)) ∧ TQ (f (n)) = TQφ (f (m)f (n)), i IQk (mn) = IQ (m) ∧ IQ (n) = IQ (f (m)) ∧ IQ (f (n)) = IQφ (f (m)f (n)), i FQk (mn) = FQ (m) ∨ IQ (n) = FQ (f (m)) ∨ FQ (f (n)) = FQφ (f (m)f (n)), i

ˇ n is seif-complementary strong SVNSGS. Every permutation φ, φ−1 on ∀mn ∈ Si , i = 1, 2, ..., n. This shows G ˇ n is strong totally self-complementary SVNSGS. Hence required {1,2,...,n} satisfies above expressions, thus G result is obtained.

Remark 2.40. Converse of theorem 2.39 may not be true, as a SVNSGS shown in Fig. 2.39 is strong totally self-complementary, it is strong and its underlying GS is a strong totally self-complementary but TQ , IQ , FQ are not constant functions.

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