Graph Structures in Bipolar Neutrosophic Environment. Received: 14 September 2017; Accepted: 27 October 2017; Published: 6 November 2017

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1 mathematics Article Graph Structures in ipolar Neutrosophic Environment Muhammad Aram, * ID, Muzzamal Sitara and Florentin Smarandache, * ID Department of Mathematics, University of the Punjab, New Campus, Lahore 54590, Paistan; muzzamalsitara@gmail.com Mathematics & Science Department, University of New Mexico, 705 Gurley Ave., Gallup, NM 8730, USA * Correspondence: m.aram@pucit.edu.p (M.A.); fsmarandache@gmail.com (F.S.); Tel.: (M.A.) Received: 4 September 07; Accepted: 7 October 07; Published: 6 November 07 Abstract: A bipolar single-valued neutrosophic (SVN) graph structure is a generalization of a bipolar fuzzy graph. In this research paper, we present certain concepts of SVN graph structures. We describe some operations on SVN graph structures and elaborate on these with examples. Moreover, we investigate some related properties of these operations. Keywords: graph structure; bipolar single-valued neutrosophic (SVN) graph structure MSC: 03E7; 05C7; 05C78; 05C99. Introduction Fuzzy graphs are mathematical models for dealing with combinatorial problems in different domains, including operations research, optimization, computer science and engineering. In 965, Zadeh [] proposed fuzzy set theory to deal with uncertainty in abundant meticulous real-life phenomena. Fuzzy set theory is affluently applicable in real-time systems consisting of information with different levels of precision. Subsequently, Atanassov [] introduced the idea of intuitionistic fuzzy sets in 986. However, for many real-life phenomena, it is necessary to deal with the implicit counter property of a particular event. Zhang [3] initiated the concept of bipolar fuzzy sets in 994. Evidently bipolar fuzzy sets and intuitionistic fuzzy sets seem to be similar, but they are completely different sets. ipolar fuzzy sets have large number of applications in image processing and spatial reasoning. ipolar fuzzy sets are more practical, advantageous and applicable in real-life phenomena. However, both bipolar fuzzy sets and intuitionistic fuzzy sets cope with incomplete information, because they do not consider indeterminate or inconsistent information that clearly appears in many systems of different fields, including belief systems and decision-support systems. Smarandache [4] 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, for which each membership value is a real standard or non-standard subset of the unit interval [0, + ]. In real-life problems, neutrosophic sets can be applied more appropriately by using the single-valued neutrosophic sets defined by Smarandache [4] and Wang et al. [5]. Deli et al. [6] considered bipolar neutrosophic sets as a generalization of bipolar fuzzy sets. They also studied some operations and applications in decision-maing problems. On the basis of Zadeh s fuzzy relations [7], Kauffman defined fuzzy graphs [8]. In 975, Rosenfeld [9] discussed a fuzzy analogue of different graph-theoretic ideas. Later on, hattacharya [0] gave some remars on fuzzy graphs in 987. Aram [] first introduced the notion of bipolar fuzzy graphs. In 0, Dinesh and Ramarishnan [] studied fuzzy graph structures and discussed their properties. In 06, Aram and Amal [3] proposed the concept of bipolar fuzzy graph structures. Certain concepts on graphs have been discussed in [4 9]. Ye [0 ] considered several applications Mathematics 07, 5, 60; doi:0.3390/math

2 Mathematics 07, 5, 60 of 0 of single-valued neutrosophic sets. Inthis research paper, we present certain concepts of bipolar single-valued neutrosophic graph structures (SVNGSs). We introduce some operations on SVNGSs and elaborate on these with examples. Moreover, we investigate some relevant and remarable properties of these operators. We have used standard definitions and terminologies in this paper. For other notations, terminologies and applications not mentioned in the paper, the readers are referred to [3 9].. ipolar Single-Valued Neutrosophic Graph Structures Definition. [4] A 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 T N, I N, F N for all v V. : V [0, + ] and there is no restriction on the sum of T N (v), I N (v) and F N (v) Definition. [5] 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 T N, I N, F N : V [0, ] and the sum of T N (v), I N (v) and F N (v) is confined between 0 and 3 for all v V. Definition 3. [3] A SVN set on a non-empty set V is an object of the form = {(v, T P(v), IP (v), FP (v), TN (v), IN (v), FN (v)) : v V} where T P, IP, FP : V [0, ] and TN, IN, FN : V [, 0]. The positive values TP (v), IP (v) and F P (v) denote the truth, indeterminacy and falsity membership values of an element v V, whereas negative values T N(v), IN (v) and FN (v) indicate the implicit counter property of truth, indeterminacy and falsity membership values of an element v V. Definition 4. [3] A SVN graph on a non-empty set V is a pair G = (, R), where is a SVN set on V and R is a SVN relation in V such that for all b, d V. TR P, d) TP ) TP, IP R, d) IP ) IP, FP R, d) FP ) FP, TR N, d) TN ) TN, IN R, d) IN ) IN, FN R, d) FN ) FN We now define the SVNGS. Definition 5. [30] A SVNGS of a graph structure Ǧ s = (V, V, V,..., V m ) is denoted by Ǧ bn = (,,,..., m ), where =< b, T P ), I P ), F P ), T N ), I N ), F N ) > is a SVN set on the set V and =<, d), T P, d), IP, d), FP, d), TN, d), IN, d), FN, d) > are the SVN sets on V such that T P, d) min{tp ), T P }, I P, d) min{ip ), I P }, F P, d) max{fp ), F P }, T N, d) max{tn ), T N }, I N, d) max{in ), I N }, F N, d) min{fn ), F N } for all b, d V. Note that 0 T P, d) + IP, d) + FP, d) 3, 3 TN, d) + IN, d) + FN, d) 0 for all, d) V, and, d) represents an edge between two vertices b and d. In this paper we use bd in place of, d).

3 Mathematics 07, 5, 60 3 of 0 Example. Consider a graph structure Ǧ s = (V, V, V, V 3 ) such that V = {b, b, b 3, b 4, b 5, b 6, b 7, b 8 }, V = {b b, b b 7, b 4 b 8, b 6 b 8, b 5 b 6, b 3 b 4 }, V = {b b 5, b 5 b 7, b 3 b 6, b 7 b 8 }, and V 3 = {b b 3, b b 4 }. Let be a SVN set on V given in Table and, and 3 be SVN sets on V, V and V 3, respectively, given in Table. Table. ipolar single-valued neutrosophic (SVN) set on vertex set V. b b b 3 b 4 b 5 b 6 b 7 b 8 T P I P F P T N I N F N Table. SVN sets, and 3. b b b b 7 b 4 b 8 b 6 b 8 b 5 b 6 b 3 b 4 b b 5 b 5 b 7 b 3 b 6 b 7 b 8 3 b b 3 b b 4 T P T P T P I P I P I P F P F P F P T N T N T N I N I N I N F N F N F N y direct calculations, it is easy to show that Ǧ bn = (,,, 3 ) is a SVNGS. This SVNGS is shown in Figure. Generated with LaTeXDraw.0.8 on Saturday March 0:30:4 PKT 07. b(0.5, 0.4, 0.6, 0.5, 0.4, 0.6) b 5 (0.3,0.,0.4, 0.3, 0., 0.4) (0.3,0.,0.6, 0.3, 0., 0.6) b 3 (0.4,0.4,0.4, 0.4, 0.4, 0.4) (0.4,0.3,0.6, 0.4, 0.3, 0.6) (0.3,0.,0.4, 0.3, 0., 0.4) 3 (0.4,0.4,0.6, 0.4, 0.4, 0.6) (0.4,0.3,0.5, 0.4, 0.3, 0.5) b (0.4,0.3,0.5, 0.4, 0.3, 0.5) (0.3,0.,0.7, 0.3, 0., 0.7) b 6 (0.4,0.4,0.7, 0.4, 0.4, 0.7) (0.3,0.4,0.7, 0.3, 0.4, 0.7) (0.3,0.4,0.5, 0.3, 0.4, 0.5) (0.4,0.4,0.7, 0.4, 0.4, 0.7) b 7 (0.5,0.5,0.4, 0.5, 0.5, 0.4) (0.4,0.4,0.6, 0.4, 0.4, 0.6) 3 (0.4,0.3,0.6, 0.4, 0.3, 0.6) (0.3,0.4,0.6, 0.3, 0.4, 0.6) b 4 (0.5,0.4,0.6, 0.5, 0.4, 0.6) b8(0.3, 0.4, 0.5, 0.3, 0.4, 0.5) Figure. A SVN graph structure. Figure.: A bipolar single-valued neutrosophic graph structure Definition 6. A SVNGS Ǧ bn = (,,,..., m ) is called a -cycle if (supp(), supp( ), supp( ),..., supp( m )) is a -cycle. finition.7. A SVNGS Ǧbn = (,,,..., m ) is called -cycle if (supp(),supp( ), p( ),...,supp( m )) is a cycle. Example. Consider a SVNGS Ǧ bn = (,, ) as shown in Figure. mple.8. Consider a SVNGS Ǧbn = (,, ) as shown in the Fig ).7, 0.9, 0.8) 0.4, 0.6, 0.6, 0.4, 0.6, 0.6) b (0.5,0.8,0.7, 0.5, 0.8, 0.7) (0.5,0.4,0.5, 0.5, 0.4, 0.5) (0.4,0.4,0.8, 0.4, 0.4, 0.8 b (0. b(0.6, 0.9, 0.8,

4 tion.7. A SVNGS Ǧbn = (,,,..., m ) is called -cycle if (supp(),supp( ), ),...,supp( m )) is a cycle. ) ple.8. Consider a SVNGS Ǧbn = (,, ) as shown in the Fig... Mathematics 07, 5, 60 4 of 0 b 3 (0.7,0.9,0.8, 0.7, 0.9, 0.8) b (0.5,0.8,0.7, 0.5, 0.8, 0.7) b 6 (0.5,0.6,0.5, 0.5, 0.6, 0.5) (0.5,0.7,0.7, 0.5, 0.7, 0.7) (0.5,0.5,0.8, 0.5, 0.5, 0.8) b 4 (0.8,0.9,0.8, 0.8, 0.9, 0.8) b 5 (0.6,0.7,0.6, 0.6, 0.7, 0.6) (0.7,0.7,0.8, 0.7, 0.7, 0.8) (0.4,0.6,0.6, 0.4, 0.6, 0.6) (0.3,0.6,0.3, 0.3, 0.6, 0.3) (0.5,0.4,0.5, 0.5, 0.4, 0.5) Figure. A SVN -cycle. (0.4,0.4,0.8, 0.4, 0.4, 0.8) (0.3,0.6,0.6, 0.3, 0.6, 0.6) Figure.: A bipolar single-valued neutrosophic -cycle Ǧ bn is a -cycle, as (supp(), supp( ),supp( )) is a -cycle, that is, b -b 3 -b 4 -b 5 -b. a -cycle as Definition (supp(),supp( 7. A SVNGS Ǧ bn = ),supp( (,,,..., )) m ) is aasvn fuzzy cycle, -cycle (for that anyis, ) if Ǧb bn b is a 3 -cycle b 4 b 5 b. and no unique -edge bd exists in Ǧ bn, such that T P d) = min{t P (e f ) : e f = supp( )}, tion.9. AI P SVNGS Ǧbn d) = min{i P (e f ) : = e f (, = supp(,,..., )}, F P d) m ) = ismax{f a SVN P (e f ) : efuzzy f = supp( -cycle )},(for any ) T -cycle and no N unique -edge bd exists in Ǧbn, such that T P d) = min{t P d) = max{t N (e f ) : e f = supp( )}, I N d) = max{i N (e f ) : e f = supp( )} (ef) : ef )} or I P or F N d) d) = min{f = min{i P N (e f ) : e f = supp( )}. (ef) : ef = supp( )} or F P d) = max{f P (ef) : ef )} or T N d) Example = 3. max{t Consider a N SVNGS (ef) : Ǧef = supp( )} or I N d) = max{i N bn = (,, ) as depicted in Figure 3. (ef) : ef )} or F N d) = min{f N (ef) : ef = supp( )}. ple.0. Consider a SVNGS Ǧbn = (,, ) as depicted in Fig..3. b (0.4,0.5,0.6, 0.4, 0.5, 0.6) (0.3,0.4,0.6, 0.3, 0.4, 0.6) (0.,0.,0.6, 0., 0., 0.6) (0.,0.5,0.5, 0., 0.5, 0.5) b (0.3,0.4,0.4, 0.3, 0.4, 0.4) (0.,0.,0.6, 0., 0., 0.6) (0.3,0.3,0.5, 0.3, 0.3, 0.5) b 4(0.3,0.3,0.5, 0.3, 0.3, 0.5) (0.,0.3,0.6, 0., 0.3, 0.6) b(0.6, 0.9, 0.8, 0.6, 0.9, 0.8) (0.4, 0.3, 0.8, 0.4, 0.3, 0.8) b 3(0.,0.5,0.6, 0., 0.5, 0.6) Figure.3: A bipolar Figuresingle-valued 3. A SVN fuzzy neutrosophic -cycle. fuzzy -cycle Ǧ bn ista P -cycle, that is, b b 4 b b 3 b and no unique -edge bd exists in Ǧbn d) = min{t P (e f ) : e f = supp( )}, I P d) = min{i P (e f ) : e f satisfying = supp( following )}, condition: F T P P d) = max{f d) min{t P P (ef) (e f ) : e f = supp( )}, T N ef = supp( )} or I P d) = max{t d) = min{i P N (e f ) : e f = supp( )}, (ef) : ef = supp( )} or I F P N d) = max{i N d) = max{f P (e f ) : e f = supp( )} or F N (ef) : ef = supp( )} or T d) = min{f 4 N N d) = max{t (e f ) : e f N = supp( )}. (ef) : ef = supp( )} or I N d) = max{i N (ef) : ef = supp( )} or F N d) = min{f N (ef) : ef = supp( )}. Definition 8. A sequence of vertices (distinct) in a SVNGS Ǧ bn = (,,,..., m ) is called a -path, Definition that is, b, b..,..., ba m, sequence such that bof vertices(distinct) b is a SVN -edge, in a SVNGS for all = Ǧbn,... =, m. (,,,..., m ) is called a -path, that is, b,b,...,b m such that b b is a bipolar singe-valued neutrosophic -edge, for all Example =,...,m. 4. Consider a SVNGS Ǧ bn = (,, ) as represented in Figure 4. Example.. Consider a SVNGS Ǧbn = (,, ) as represented in the Fig..4. b b (0.3,0.5,0.4, 0.3, 0.5, 0.4) 0.5, 0.6) (0.,0.3,

5 m =,...,m. - positive strength of connectedness of truth between two nodes b and d is defined by: P Pl Pl Pl P Example.. Consider a SVNGS Ǧbn = (,, ) as represented in the Fig..4. Mathematics 07, 5, 60 5 of 0 b(0.4, 0.6, 0.6, 0.4, 0.6, 0.6) (0.3,0.5,0.6, 0.3, 0.5, 0.6) (0.,0.3,0.6, 0., 0.3, 0.6) b (0.3,0.5,0.4, 0.3, 0.5, 0.4) (0.,0.3,0.6, 0., 0.3, 0.6) (0.,0.6,0.5, 0., 0.6, 0.5) b 4(0.3,0.4,0.5, 0.3, 0.4, 0.5) (0.,0.4,0.6, 0., 0.4, 0.6) b3(0., 0.6, 0.6, 0., 0.6, 0.6) Figure 4. A SVN -path. In In this this SVNGS SVNGS, sequence the Figure sequence of.4: distinct of Adistinct bipolar vertices vertices single-valued b b, b 4, bneutrosophic 3, b is a SVN -path.,b 4,b 3,b is a SVN -path. Definition Let Let Ǧ bn Ǧbn = (, = (,,,..., m ) be a SVNGS. The positive truth strength T P,,..., m ) be a SVNGS. Positive truth strength.p T, P positive.p, positive falsity falsity strength strength F P.P 5 F, and P.Ppositive indeterminacy strength I P, and positive indeterminacy strength.p of a I P -path,.p Pof a= b -path,, b,... P, b n, are = bdefined,b,...,b as n are defined as; T P.P = n [T P h b h )], I P.P n [I P h b h )], F P.P = n [F P h b h )] T P.P = n [T P h b h )], I P.P = n [I P h b h )], F P.P = n [F P h b h )]. h= h= h= h= h= h= Similarly, the negative truth strength T N.P, negative falsity strength F N.P, and negative indeterminacy Similarly, negative strength I N truth strength T N.P, negative falsity strength F N.P, and negative indeterminacy.p of a -path are defined as strength I N.P of a -path, are defined as; T N.P = n [T N h b h )], I N.P [I h b h )], F N.P = n T N.P [F N h b h )] = n [T N h b h )], I N.P = n [I N h b h )], F N.P = n [F N h b h )]. h= h= h= h= h= h= Example Consider Consider a SVNGS a SVNGS Ǧ bn = Ǧbn (, =,(,, 3 ), as shown, 3 ) as inshown Figurein 5. the Fig..5. b5(0., 0., 0.3, 0., 0., 0.3) (0., 0., 0.5, 0., 0., 0.5) b 6(0.3,0.4,0.6, 0.3, 0.4, 0.6) (0.,0.,0.6, 0., 0., 0.6) b 3(0.3,0.4,0.3, 0.3, 0.4, 0.3) b (0.4,0.4,0.5, 0.4, 0.4, 0.5) (0.3, 0.4, 0.6, 0.3, 0.4, 0.6) (0.,0.,0.3, 0., 0., 0.3) 3(0.3,0.4,0.5, 0.3, 0.4, 0.5) (0.,0.4,0.6, 0., 0.4, 0.6) (0.3,0.3,0.5, 0.3, 0.3, 0.5) b 7(0.4,0.5,0.3, 0.4, 0.5, 0.3) b (0.3,0.3,0.4, 0.3, 0.3, 0.4) b 8(0.,0.4,0.4, 0., 0.4, 0.4) (0.,0.4,0.4, 0., 0.4, 0.4) (0.3, 0.3, 0.4, 0.3, 0.3, 0.4) (0.3,0.4,0.5, 0.3, 0.4, 0.5) (0., 0.3, 0.5, 0., 0.3, 0.5) 3(0.3,0.3,0.5, 0.3, 0.3, 0.5) b4(0.4, 0.4, 0.5, 0.4, 0.4, 0.5) Figure 5. A bipolar single-valued Figure.5: neutrosophic A SVNGSgraph Ǧbn = structure (, (SVNGS),, 3 ) Ǧ bn = (,,, 3 ). In this SVNGS there is a -path, that is, P = b 5,b 6,b 8,b 4,b 3. So, T N.P = 0., I N.P = 0., F N.P = 0.6, T P.P = 0., I P.P = 0. and F P.P = 0.6 Definition.5. Let Ǧbn = (,,,..., m ) be a SVNGS. Then

6 Mathematics 07, 5, 60 6 of 0 In this SVNGS, there is a -path, that is, P = b 5, b 6, b 8,b 4, b 3. Thus, T N.P = 0., I N.P = 0., F N.P = 0.6, T P.P = 0., I P.P = 0. and F P.P = 0.6. Definition 0. Let Ǧ bn = (,,,..., m ) be a SVNGS. Then The -positive strength of connectedness of truth between two nodes b and d is defined by d) = {T Pl d)}, such that T Pl d) = (T Pl T P )d) for l T P l and T P d) = (T P T P )d) = (T P y y) T P (yd)). The -positive strength of connectedness of indeterminacy between two nodes b and d is defined by I P d) = {I Pl l d)}, such that I Pl d) = (I Pl I P )d) for l and I P d) = (I P I P )d) = (I P y y) I P (yd)). The -positive strength of connectedness of falsity between two nodes b and d is defined by F P d) = {F Pl l d)}, such that F Pl d) = (F Pl F P )d) for l and F P d) = (F P F P )d) = (F P y y) F P (yd)). The -negative strength of connectedness of truth between two nodes b and d is defined by T N d) = {T Nl l d)}, such that T Nl d) = (T Nl T N )d) for l and T N d) = (T N T N )d) = (T N y y) T N (yd)). The -negative strength of connectedness of indeterminacy between two nodes b and d is defined by I N d) = {I Nl l d)}, such that I Nl d) = (I Nl I N )d) for l and I N d) = (I N I N )d) = (I N y y) I N (yd)). l The -negative strength of connectedness of falsity between two nodes b and d is defined by F N and F N d) = {F Nl l d) = (F N d)}, such that F Nl d) = (F Nl F N )d) = y (F N y) F N (yd)). F N )d) for l Definition. Let Ǧ bn = (,,,..., m ) be a SVNGS and b be a node in Ǧ bn. Let (,,,..., m) be a SVN subgraph structure of Ǧ bn induced by \ {b} such that e = b, f = b F P T P ) = I P ) = F P ) = T P e) = I P e) = F P e) = 0 T N ) = I N ) = F N ) = T N e) = I N e) = F N e) = 0 T P (e) = T P(e), IP (e) = I P(e), FP (e) = F P(e), TN (e) = T N(e), IN (e) = I N(e), FN e) = F N(e) T P (e f ) = T P (e f ), I P (e f ) = I P (e f ), F P (e f ) = F P (e f ), T N (e f ) = T N (e f ), I N (e f ) = I N (e f ) Then b is a SVN fuzzy cut-vertex if T P (e f ), T N (e f ) < T N (e f ), I N F N (e f ) = F N (e f ), edges be, e f Ǧ bn (e f ) < I N (e f ) > T P (e f ) and F N Note that vertex b is a SVN fuzzy T P cut-vertex if T P cut-vertex if I P (e f ) > I P (e f ), I P (e f ) < F N (e f ) > T P (e f ), and it is a SVN fuzzy F P cut-vertex if F P Moreover, vertex b is a SVN fuzzy T N cut-vertex if T N cut-vertex if I N (e f ) < I N (e f ) < T N (e f ) and it is a SVN fuzzy F N cut-vertex if F N (e f ) > I P (e f ), F P (e f ) > (e f ), for some e, f \ {b}. (e f ), it is a SVN fuzzy I P (e f ) > F P (e f ). (e f ), it is a SVN fuzzy I N (e f ) < F N (e f ). Example 6. Consider a SVNGS Ǧ bn = (,, ) as depicted in Figure 6, and let Ǧ bh = (,, ) be a SVN subgraph structure of the SVNGS Ǧ bn, which is obtained through deletion of vertex b.

7 Mathematics 07, 5, 60 7 of 0 b5(0.4, 0.5, 0.6, 0.4, 0.5, 0.6) b (0.4,0.7,0.5, 0.4, 0.7, 0.5) (0.4,0.4,0.5, 0.4, 0.4, 0.5) (0.3,0.5,0.4, 0.3, 0.5, 0.4) (0.3,0.6,0.4, 0.3, 0.6, 0.4) (0.,0.4,0., 0., 0.4, 0.) (0.3,0.3,0.4, 0.3, 0.3, 0.4) (0., 0.4, 0., 0., 0.4, 0.) b (0.3,0.6,0.4, 0.3, 0.6, 0.4) (0.3,0.,0.4, 0.3, 0., 0.4) b6(0.3, 0.4, 0.4, 0.3, 0.4, 0.4) (0., 0.4, 0.3, 0., 0.4, 0.3) (0.5,0.7,0.5, 0.5, 0.7, 0.5) b 4 (0.5,0.7,0.5, 0.5, 0.7, 0.5) b3(0.5, 0.7, 0.5, 0.5, 0.7, 0.5) Figure 6. A SVNGS Ǧ bn = (,, ). Figure.6: A SVNGS Ǧbn = (,, ) The vertex b is a SVN fuzzy The vertex b is a SVN fuzzy I P I P cut-vertex and a SVN fuzzy cut vertex and SVN fuzzy I N I N cut-vertex, cut vertex, since I P because I P b 5 ) = 0, I P b 5 ) = 0, I P b 5 ) = 0.5, I P b 5 ) = 0.5, I P 4 b 3 ) = b 3 ) =, I P 0.7, I P 4 b 3 ) = 4 b 3 ) 0.7 and I P 0.7, I P 3 b 5 ) = 0.3, 3 b 5 ) = 0.3, I P 3 b 5 ) = I 0.4. I N P 3 b 5 ) = 0.4. I N b 5 ) = 0 > 0.5 b 5 ) = 0 > 0.5 = I = I N N b 5 ), I N b 5 ), I N 4 b 3 ) 4 b 3 ) = 0.7 = I = 0.7 = I N N 4 b 3 ) and I N 4 b 3 ) and I N 3 b 5 ) = 0.3 > 0.4 = I 3 b 5 ) = 0.3 > 0.4 = I N N 3 b 5 ). 3 b 5 ) Definition. Suppose Ǧ bn = (,,,..., m ) is a SVNGS and bd is a -edge. Let (, Definition.8. Suppose Ǧbn = (,,,..., m ) is a SVNGS and bd is, a,..., m) be a SVN fuzzy spanning subgraph structure of Ǧ bn, such that edge. Let (,,,..., m ) be a SVN fuzzy spanning subgraph-structure of Ǧbn, such that T P T P d) = 0 = I P d) = 0 = I P d) = F P d) = F P d), T N d) = 0 = I N d), T N d) = F N d) = 0 = I N d) T P d) = F N d), T P (gh) = T P (gh), I (gh) = I P (gh) = T P (gh), I P (gh), F P = I (gh) P (gh) F P d) F P d), T N (gh) = T N (gh), I N = I (gh) N (gh), d) = F P d), T N (gh) = T N (gh), I N (gh) = I N (gh), F N F N = F d) N = F d) N d), edges gh = bd d), edges gh bd. Then bd is a SVN fuzzy -bridge if T P (e f ) > T P (e f ), I P (e f ) > I P (e f ), F P (e f ) > Then bd is F P a SVN fuzzy -bridge if T P (e f ), T (ef) > T P N (e f ) < T N (ef),i P (e f ), I (ef) > I P N (e f ) < I N (e f ) and F (ef)andf P N (e f ) < F N (ef) > F P (e f ), for some e, f V. Note that bd is a SVN fuzzy (ef), T N (ef) < T N (ef), I N T P bridge if T (ef) < I N (ef) and F N (ef) < F N P (e f ) > T P (e f ), it is a SVN fuzzy I P bridge if I (ef), for some e,f V. P (e f ) > I P f ) and it is a SVN fuzzy F P bridge if F P (e f ) > F P (e f ). Moreover, bd is a SVN Note that bd is a SVN fuzzy T P bridge if T P (ef) > T P fuzzy (ef), SVN fuzzy I P T N bridge if T bridge if N (e f ) < T N (e f ), it is a SVN fuzzy I N bridge if I N (e f I P (ef) > I P ) < I N (e f ) (ef) and SVN fuzzy F P bridge if F P (ef) > F P and it is a SVN fuzzy F N bridge if F N (e f ) < F N (e f ). (ef). Moreover, bd is a SVN fuzzy T N bridge if T N (ef) < T N (ef), SVN fuzzy I N bridge if I N Example (ef) < I N 7. Consider a SVNGS Ǧ bn = (,, (ef) and SVN fuzzy F N ) as depicted in bridge if F N Figure 6 and Ǧ (ef) < F N bs = (ef). (,, ), a SVN spanning subgraph structure of the SVNGS Ǧ bn obtained by deleting -edge b 5 ) and that is shown in Figure 7. Example.9. Consider a SVNGS Ǧbn = (,, ) as depicted in Fig..6 and Ǧ bs = (,, ) is SVN spanning subgraph-structure of SVNGS Ǧbn obtained by deleting -edge b 5 ) and shown in Fig..7.

8 Mathematics 07, 5, 60 8 of 0 b (0.4,0.7,0.5, 0.4, 0.7, 0.5) b6(0.3, 0.4, 0.4, 0.3, 0.4, 0.4) b6(0.3, 0.4, 0.4, 0.3, 0.4, 0.4) (0.,0.4,0., 0., 0.4, 0.) (0.3,0.6,0.4, 0.3, 0.6, 0.4) b5(0.4, 0.5, 0.6, 0.4, 0.5, 0.6) (0.3,0.5,0.4, 0.3, 0.5, 0.4) (0.,0.4,0., 0., 0.4, 0.) (0.3,0.3,0.4, 0.3, 0.3, 0.4) b 4 (0.5,0.7,0.5, 0.5, 0.7, 0.5) (0.,0.4,0., 0., 0.4, 0.) (0.,0.4,0., 0., 0.4, 0.) (0.,0.4,0.3, 0., 0.4, 0.3) (0.3,0.,0.4, 0.3, 0., 0.4) (0.5,0.7,0.5, 0.5, 0.7, 0.5) s edge b 5 ) is a SVN fuzzy -bridge. Figure 7. AAs SVNGS T Ǧ bs = (, P Figure.7: A SVNGS b 5 ), = Ǧ bs = ( ). 0.3, T,, ) P b 5 ) = 0.4, I P b 5 )= 0 b 5 ) = 0.4, F This P edge b 5 )= 0.4, F b 5 ) is a SVN P fuzzy b -bridge, 5 ) = 0.5. And T as T P This edge b 5 ) is a SVN fuzzy -bridge. As T P b 5 ) = 0.3, N T P b b 5 ) = 0.3 > 0.4 = T 5 0.4, I P 0.3, T P b 5 )= 0.3, N b b 5 )= 0.3 I P > b 5 ) = 0.4, I P b 5 )= 0.3, I P b 0.4= 5 ) = 0.4, FI P b 5 ) = 0.4, F P N b 5 )= 0.4, F P b 5 )= 0.4, F P b 5 ) = 0.5., T N b 5 ) = 0.3 > 0.4 = T b 5 ) = 0.5. And T N N b 5 ), I N b 5 ), F N b 5 )= 0.4 > 0.5 = F N b 5 )= b 5 ). 0.3 > 0.4= I N b 5 ) = 0.3 > 0.4 = T N b 5 ), I N b 5 ), and F N b 5 )= 0.3 > 0.4= I N b 5 )= 0.4 > 0.5 = F N b 5 ), F N b 5 ). finition.0. A SVNGS Ǧbn = (,, b,..., 5 )= 0.4 m ) > is 0.5 a = -tree, F N if (supp(),supp( b 5 ). ),supp( supp( m )) is Definition a tree. 3. A SVNGS Alternatively, Ǧ bn = (, Ǧbn, is,.. a., m )-tree, is a -tree if Ǧbn if (supp(), has asubgraphinduced supp( ), supp( ), by supp( t forms Definition tree , supp(a m )) SVNGS is a -tree. Ǧbn Alternatively, = (,, Ǧ bn,..., is a -tree m ) is if Ǧa bn has -tree, a subgraph if (supp(),supp( induced by ),supp( ) ),...,supp(that m )) forms is atree. tree. Alternatively, Ǧbn is a -tree, if Ǧbn has asubgraphinduced by supp( ), mple that.. forms tree. Consider the SVNGS Example 8. Consider the SVNGS Ǧ Ǧbn = (,, ) as depicted in Fig..8 bn = (,, ) as depicted in Figure 8. Example.. Consider the SVNGS Ǧbn = (,, ) as depicted in Fig..8 (0.4, 0.3, 0.8, 0.4, 0.3, 0.8) b (0.6,0.9,0.8, 0.6, 0.9, 0.8) (0.3,0.6,0.6, 0.3, 0.6, 0.6) (0.3,0.6,0.6, 0.3, 0.6, 0.6) b 5(0.6,0.7,0.6, 0.6, 0.7, 0.6) (0.4,0.4,0.8, 0.4, 0.4, 0.8) (0.5,0.5,0.8, 0.5, 0.5, 0.8) (0.5,0.7,0.7, 0.5, 0.7, 0.7) b (0.5,0.8,0.7, 0.5, 0.8, 0.7) b 4(0.8,0.9,0.8, 0.8, 0.9, 0.8) (0.3,0.6,0.3, 0.3, 0.6, 0.3) (0.3,0.6,0.3, 0.3, 0.6, 0.3) b 6(0.5,0.6,0.5, 0.5, 0.6, 0.5) (0.5,0.4,0.5, 0.5, 0.4, 0.5) b3(0.5, 0.7, 0.5, 0.5, 0.7, 0.5) b(0.3, 0.6, 0.4, 0.3, 0.6, 0.4) (0.7,0.7,0.8, 0.7, 0.7, 0.8) (0.4,0.6,0.6, 0.4, 0.6, 0.6) b 3(0.7,0.9,0.8, 0.7, 0.9, 0.8) Figure Figure.8:.8: A A bipolar bipolar single-valued neutrosophic neutrosophic -tree -tree This SVNGS Ǧ This SVNGS Ǧbn = (, bn = (,,, ) ) is a is a -tree, as (supp(), supp( -tree since (supp(),supp( ), supp( ),supp( )) is a -tree. )) is a tree. This SVNGS Ǧbn = (,, ) is a -tree since (supp(),supp( ),supp( )) is a tree. Definition 4. A SVNGS Definition.. A SVNGS Ǧbn Ǧ bn = (,,,,...,,..., m ) is a SVN fuzzy m ) is a SVN -tree if Ǧ fuzzy bn has a SVN -tree, if Ǧbn fuzzy has a SVN spanning structure finition fuzzy.. spanning Asubgraph-structure SVNGS Ǧbn Ȟ = bn Ȟbn = (, =, (,,,..., m),,..,...,., m) such that for all m ) is asuch SVN -edges, bd not in Ȟ that for fuzzy all -edges bn : -tree, bd ifnot Ǧbninhas Ȟbn, a SV y spanning subgraph-structure. Ȟ. Ȟ bn is a -tree Ȟbn = (,,,..., m) bn is a such that for all -edges bd not in Ȟ -tree.. Ȟ bn. ista P d) -tree < T P d), I P d) < I P I N. T P d) < T P (0.4, 0.3, 0.8, 0.4, 0.3, 0.8) b (0.6,0.9,0.8, 0.6, 0.9, 0.8) b (0.4,0.7,0.5, 0.4, 0.7, 0.5) b5(0.4, 0.5, 0.6, 0.4, 0.5, 0.6) (0.3,0.5,0.4, 0.3, 0.5, 0.4) d), F N d) > F N d). d), I P d) < I P N N N (0.3,0.6,0.4, 0.3, 0.6, 0.4) (0.3,0.3,0.4, 0.3, 0.3, 0.4) b 4 (0.5,0.7,0.5, 0.5, 0.7, 0.5) d), F P d) < F P (0.,0.4,0.3, 0., 0.4, 0.3) (0.3,0.,0.4, 0.3, 0., 0.4) (0.5,0.7,0.5, 0.5, 0.7, 0.5) Figure.7: A SVNGS Ǧ bs = (,, ) b 5(0.6,0.7,0.6, 0.6, 0.7, 0.6) (0.5,0.5,0.8, 0.5, 0.5, 0.8) (0.5,0.7,0.7, 0.5, 0.7, 0.7) b (0.5,0.8,0.7, 0.5, 0.8, 0.7) Figure 8. A SVN -tree. b 4(0.8,0.9,0.8, 0.8, 0.9, 0.8) b 6(0.5,0.6,0.5, 0.5, 0.6, 0.5) d), F P d) < F P b3(0.5, 0.7, 0.5, 0.5, 0.7, 0.5) (0.3, 0.6, 0.4, 0.3, 0.6, 0.4) (0.7,0.7,0.8, 0.7, 0.7, 0.8) b 3(0.7,0.9,0.8, 0.7, 0.9, 0.8) d), T N d) > T N d), I N d) > d), T N d) > T N d), I N d)

9 Mathematics 07, 5, 60 9 of 0. T P d) < T P I N d) > I N d), I P d) < I P d), and F N d) > F N d). d), F P d) < F P d), T N d) > T N d), In particular, Ǧ bn is a SVN fuzzy T P tree if T P d) < T P d), it is a SVN fuzzy I P tree Inparticular, if I P d) < Ǧbn I P d), is a SVN and it is fuzzy a SVN T fuzzy P tree if F P T P tree if F P d) > F P d) < T P d), a d). SVN Moreover, fuzzy Ǧ I bn is P a tree SVN if Ifuzzy P d) < I P T N d) and tree if T N a SVN fuzzy d) > T N d), F it is a P tree if F SVN fuzzy P I N tree if I N d) > I N d) > F P d). d), and it is a Moreover, SVN fuzzy Ǧbn is a SVN F N fuzzy tree if F N T N d) < F N tree if T N d). d) > T N d), a SVN fuzzy I N tree if I N d) > I N d) and a SVN fuzzy F N tree if F N d) < F N d). Example Consider Consider the SVNGS the SVNGS Ǧ bn = Ǧbn (, =,(, ) as, depicted ) as in depicted Figure 9. in Fig..9. (0.,0.,0.5, 0., 0., 0.5) (0.3, 0.3, 0.4, 0.3, 0.3, 0.4) (0.,0.6,0., 0., 0.6, 0.) b 5(0.4,0.5,0.6, 0.4, 0.5, 0.6) (0.,0.4,0.4, 0., 0.4, 0.4) (0.,0.4,0., 0., 0.4, 0.) b 4(0.5,0.7,0.5, 0.5, 0.7, 0.5) b (0.4,0.7,0.5, 0.4, 0.7, 0.5) b (0.3,0.6,0.5, 0.3, 0.6, 0.5) (0.3, 0.5, 0.4, 0.3, 0.5, 0.4) (0.3,0.,0.3, 0.3, 0., 0.3) b 6(0.3,0.4,0.4, 0.3, 0.4, 0.4) b 3(0.5,0.7,0.5, 0.5, 0.7, 0.5) (0.5,0.5,0.4, 0.5, 0.5, 0.4) (0.,0.4,0.4, 0., 0.4, 0.4) Figure 9. A SVN fuzzy Figure.9: SVN fuzzy -tree. -tree It Itis -tree, not rather -tree. thanut a -tree. it is SVN However, fuzzy itis a-tree, SVNsince fuzzy it has -tree, a SVN because fuzzy it has spanning a SVN subgraph fuzzy ( spanning,, ) subgraph as a ( -tree,, which is obtained through deletion of -edge b b 5 from Ǧbn. Moreover, T P b 5 ) = 0.3, T P, ) as a b 5 ) = 0., I P -tree, which is obtained through the deletion of the -edge b b 5 from Ǧ b 5 ) = 0.3, I P b 5 ) 0. and F P bn. Moreover, T P b 5 ) = 0.4, F b 5 ) = 0.5. T N b 5 ) = 0.3, T P b 5 ) = 0.3 < 0. = T N b 5 ) = 0., I P b 5 ), I N b 5 ) = 0.3, I P b 5 ) = 0., F P b 5 ) = 0.3 < 0. = I N b 5 ) = 0.4, F b 5 ) and F N b 5 ) = b 5 ) = 0.5. T N 0.4 > 0.5 = F b 5 ) = 0.3 < 0. = T N b 5 ). b 5 ), I N b 5 ) = 0.3 < 0. = I N b 5 ) and F N b 5 ) = 0.4 > 0.5 = F b 5 ). Now we define the operations on SVNGSs. Definition.4. Now we define LetǦb the operations =(,, on SVNGSs.,..., m )andǧb =(,,,..., m )betwosvngss. Cartesian product of Ǧb and Ǧb, denoted by Definition 5. Let Ǧ b b = ( Ǧb, = (,,...,, m ) and, Ǧ b = (,...,,,,. m.., m m ), ) be two SVNGSs. The Cartesian product of Ǧ b and Ǧ b, denoted by is defined as: T( P d) = Ǧ b ) (TP Ǧ T b = P ( )d) = T, P ) T, P,..., m m ) (i) I( P ) d) = (IP I P )d) = I P ) I P is defined as F( P d) = ) (FP F P )d) = F P ) F P T T( N ( P d) ) d) = = (TP ) (TN T T N P )d) )d) = T T N P ) T ) T N P (i) (ii) I( N I( P ) d) ) d) = = (IP (IN I N I P )d) = )d) = I N I P ) ) I N P F( N F( P d) = ) (FN F N )d) N ) F ) d) = (FP F P )d) = F P ) F P for all d) V V, 0

10 d3(0.5, 0.3, 0.5, 0.5, 0.3, 0.5) Mathematics 07, 5, 60 0 of 0 T N ( ) d) = (TN T N )d) = T N ) T N (ii) I N ( ) d) = (IN I N )d) = I N ) I N F N ( ) d) = (FN F N )d) = F N ) F P T( P ) for all d) d )d ) = (T P T P )d )d ) = T P ) T P (d d ) V V, (iii) I( P T( P ) d )d ) = (I ) d P )d ) = I (T P P )d )d ) = I T P P ) I )d )d T P ) (d d ) T P (d d ) F( P (iii) I( P ) d )d ) = (F ) d P F )d ) = (I P P I P )d )d ) = F )d )d ) = I P P ) F ) I P P (d d ) (d d ) T F( P ) )d ) = (F P F P F P F P (d d ) ( N ) d )d ) = (T N T N )d )d ) = T N ) T N (d d ) (iv) I( N T N ( ) )d ) = (T N T N )d )d ) = T N T N ) d )d ) = (I N I N )d )d ) = I N ) I N (d d ) (iv) F( N I N ( ) )d ) = (I N I ) I N ) I N ) d )d ) = (F N F N )d )d ) = F N ) F(d N d(d ) d ) for all b FV N, (d ( ) d d ) )d V ), = (F N F N )d )d ) = F N ) F N (d d ) for all b V, (d d ) V, and T( P T( P ) d) ) d) = (T P d) d) = (T T P P T ) P d) d) = T P ) d) d) = T P T T P P b ) (v) I( P (v) I( P ) d) ) d) = (I P d) d) = I (I P P I ) d) d) = I P ) d) d) I P I I P P b ) b ) F( P F( P ) d) d) = (F ) P d) d) = F (F P P ) d) d) = F F P ) d) d) = F P P F F P P b ) b ) T( N T N ( ) d) d) = (T N T N ) T ) d) d) = (T N T N ) d) d) = T N T N b b) ) (vi) (vi) I( N I N ( ) d) d) = (I N I N ) ) b ) d) = (I N I N ) d) d) = I N I N b ) F( N F N ( ) d) d) = (F N F N F N ) d) = (F N F N ) d) d) = F N F N b ) b ) for all for d allv d, V b, ) b V). V. Example Example Consider Consider Ǧ Ǧb b = ( =,(,, ), and Ǧ) and b = ( Ǧb,, = ( ) as, SVNGSs, ) ofare GSsSVNGSs Ǧ s = (V, Vof, VGSs ) Ǧs = (V,V and Ǧ,V s = ) (V and, V Ǧs, V = ), (V respectively,,v,v ), as respectively, depicted in Figure as depicted 0, where in V Fig. = {b.0, b }, where V = V {b = 3 b 4 }, {b b }, V = V {b 3 b = 4 }, {d V d }, = and {d V d }, = V {d d = 3 }. {d d 3 }. (0.5,0.,0.8, 0.5, 0., 0.8) (0.4, 0., 0.6, 0.4, 0., 0.6) b 4(0.5,0.,0.6, 0.5, 0., 0.6) b (0.5,0.,0.8, 0.5, 0., 0.8) b3(0.4, 0., 0.4, 0.4, 0., 0.4) b(0.5, 0., 0.6, 0.5, 0., 0.6) d(0., 0., 0.3, 0., 0., 0.3) (0., 0., 0.4, 0., 0., 0.4) (0.3,0.,0.5, 0.3, 0., 0.5) d (0.3,0.,0.4, 0.3, 0., 0.4) Figure 0. Two SVNGSs Ǧ Figure.0: Two SVNGSs b Ǧb and Ǧ and b. Ǧb The Cartesian Cartesian product of Ǧb product of and Ǧb Ǧ b and Ǧ defined b, defined as Ǧb as Ǧ b Ǧ Ǧb = b = { {,,,, }, is depicted } is depicted in Fig. in. Figures and.. and.

11 Mathematics 07, 5, 60 of 0 b d 3(0.5,0.,0.8, 0.5, 0., 0.8) b d (0.3,0.,0.6, 0.3, 0., 0.6) b d 3(0.5,0.,0.8, 0.5, 0., 0.8) (0.5,0.,0.8, 0.5, 0., 0.8) (0.5,0.,0.8, 0.5, 0., 0.8) b d (0.3,0.,0.6, 0.3, 0., 0.6) b d (0.,0.,0.8, 0., 0., 0.8) (0.3,0.,0.8, 0.3, 0., 0.8) (0.,0.,0.6, 0., 0., 0.6) b d (0.,0.,0.8, 0., 0., 0.8) (0.3,0.,0.8, 0.3, 0., 0.8) (0.,0.,0.6, 0., 0., 0.6) (0.3,0.,0.6, 0.3, 0., 0.6) (0.3,0.,0.6, 0.3, 0., 0.6) (0.,0.,0.8, 0., 0., 0.8) b d (0.,0.,0.6, 0., 0., 0.6) (0.3,0.,0.8, 0.3, 0., 0.8) b d (0.,0.,0.6, 0., 0., 0.6) Figure.: Ǧ b Ǧb (0.3,0.,0.8, 0.3, 0., 0.8) (0.,0.,0.8, 0., 0., 0.8) b d 3(0.5,0.,0.6, 0.5, 0., 0.6) (0.,0.,0.8, 0., 0., 0.8) (0.,0.,0.8, 0., 0., 0.8) b d 3(0.5,0.,0.6, 0.5, 0., 0.6) b d (0.3,0.,0.8, 0.3, 0., 0.8) b d (0.3,0.,0.8, 0.3, 0., 0.8) (0.5,0.,0.6, 0.5, 0., 0.6) (0.5,0.,0.6, 0.5, 0., 0.6) b 3d (0.3,0.,0.4, 0.3, 0., 0.4) b 4d 3(0.5,0.,0.6, 0.5, 0., 0.6) b 3d (0.3,0.,0.4, 0.3, 0., 0.4) b 4d 3(0.5,0.,0.6, 0.5, 0., 0.6) Figure.: Ǧ b Ǧb b 4d (0.,0.,0.6, 0., 0., 0.6) (0.3,0.,0.6, 0.3, 0., 0.6) (0.,0.,0.4, 0., 0., 0.4) Figure. Ǧ b Ǧ b. b 4d (0.,0.,0.6, 0., 0., 0.6) (0.3,0.,0.6, 0.3, 0., 0.6) (0.,0.,0.4, 0., 0., 0.4) (0.3,0.,0.5, 0.3, 0., 0.5) (0.3,0.,0.5, 0.3, 0., 0.5) (0.,0.,0.6, 0., 0., 0.6) b 3d (0.,0.,0.4, 0., 0., 0.4) (0.3,0.,0.6, 0.3, 0., 0.6) b 3dFigure (0.,0.,0.4, 0., 0., 0.4). Ǧ b Ǧ b. (0.3,0.,0.6, 0.3, 0., 0.6) (0.,0.,0.6, 0., 0., 0.6) b 3d 3(0.4,0.,0.5, 0.4, 0., 0.5) (0.,0.,0.6, 0., 0., 0.6) (0.,0.,0.6, 0., 0., 0.6) b 3d 3(0.4,0.,0.5, 0.4, 0., 0.5) b 4d (0.3,0.,0.6, 0.3, 0., 0.6) Figure.: Ǧ b Ǧb Theorem. The Cartesian product Ǧ b Ǧ b = (,,,..., m m ) of two Theorem SVNSGSs.6. of GSs Cartesian Gˇ s and Gproduct ˇ s Ǧb Ǧb = (,,,..., m m ) of two SVNSGSs of GSs Gˇ s and G ˇ is a SVNGS of Gˇ s Gˇ s. s is a SVNGS of Gˇ s G ˇ s. Proof. Consider two cases: Figure.: Ǧ b Ǧb Proof. Consider two cases: b 4d (0.3,0.,0.6, 0.3, 0., 0.6) Theorem Case Case.. For For.6. b Cartesian V product Ǧb Ǧb, d d V, = (,,,..., m m ) of two SVNSGSs of GSs Gˇ s and G ˇ s is a SVNGS of Gˇ s G ˇ s. T ( P Proof. Consider two cases: ) ) T ) (d )d )) = T P T P (d (d d ) T P ) [T ) T P (d ) T P Case. For b V, d d V = [T ) T )] [T ) T P T P (d P T P T( P ) (d )d )) = T ) T P (d d ) ( ) ) T( d ) ), ( P T P ) d ) T ) [T P ( P ) d ) (d ) T P [T P ) T P (d )] [T P ) T P T N ( ) (d )d )) = T N = T( P ) d T N (d d ) ) ) T( P d ) ), T N ) [T N (d ) T N = [T N ) T N (d )] [T N ) T N = T N ( ) d ) T N ( ) d )

12 Mathematics 07, 5, 60 of 0 I( P ) (d )d )) = I P ) I P (d d ) I P ) [I P (d ) I P = [I P ) I P (d )] [I P ) I P = I( P ) d ) I( P ) d ) I N ( ) (d )d )) = I N ) I N (d d ) I N ) [I N (d ) I N = [I N ) I N (d )] [I N ) I N = I N ( ) d ) I N ( ) d ) F( P ) (d )d )) = F P ) F P (d d ) F P ) [F P (d ) F P = [F P ) F P (d )] [F P ) F P = F( P ) d ) F( P ) d ) F N ( ) (d )d )) = F N ) F N (d d ) F N ) [F N (d ) F N = [F N ) F N (d )] [F N ) F N = F N ( ) d ) F N ( ) d ) for bd, bd V V. Case. For b V, d d V, T( P ) ((d b)(d b)) = T P ) T P (d d ) T P ) [T P (d ) T P = [T P ) T P (d )] [T P ) T P = T( P ) (d b) T( P ) (d b) T N ( ) ((d b)(d b)) = T N ) T N (d d ) T N ) [T N (d ) T N = [T N ) T N (d )] [T N ) T N = T N ( ) (d b) T N ( ) (d b) I( P ) ((d b)(d b)) = I P ) I P (d d ) I P ) [I P (d ) I P = [I P ) I P (d )] [I P ) I P = I( P ) (d b) I( P ) (d b)

13 Mathematics 07, 5, 60 3 of 0 I N ( ) ((d b)(d b)) = I N ) I N (d d ) I N ) [I N (d ) I N = [I N ) I N (d )] [I N ) I N = I N ( ) (d b) I N ( ) (d b) F( P ) ((d b)(d b)) = F P ) F P (d d ) F P ) [F P (d ) F P = [F P ) F P (d )] [F P ) F P = F( P ) (d b) F( P ) (d b) for d b, d b V V. F N ( ) ((d b)(d b)) = F N ) F N (d d ) F N ) [F N (d ) F N = [F N ) F N (d )] [F N ) F N = F N ( ) (d b) F P ( ) (d b) oth cases hold for all {,,..., m}. This completes the proof. Definition 6. Let Ǧ b = (,,,..., m ) and Ǧ b = (,,,..., m ) be two SVNGSs. The cross product of Ǧ b and Ǧ b, denoted by Ǧ b Ǧ b = (,,,..., m m ) is defined as T( P ) d) = (TP T P )d) = T P ) T P (i) I( P ) d) = (IP I P )d) = I P ) I P F( P ) d) = (FP F P )d) = F P ) F P T N ( ) d) = (TN T N )d) = T N ) T N (ii) I N ( ) d) = (IN I N )d) = I N ) I N F N ( ) d) = (FN F N )d) = F N ) F P (iii) (iv) for all d) V V, and T( P ) d ) d ) = (T P T P ) d ) d ) = T P b ) T P (d d ) I( P ) d ) d ) = (I P I P ) d ) d ) = I P b ) I P (d d ) F( P ) d ) d ) = (F P F P ) d ) d ) = F P b ) F P (d d ) T N ( ) d ) d ) = (T N T N ) d ) d ) = T N b ) T N (d d ) I N ( ) d ) d ) = (I N I N ) d ) d ) = I N b ) I N (d d ) F N ( ) d ) d ) = (F N F N ) d ) d ) = F N b ) F N (d d ) for all b ) V, (d d ) V. Example. The cross product of SVNGSs Ǧ b and Ǧ b shown in Figure 0 is defined as Ǧ b Ǧ b = {,, } and is depicted in Figure 3.

14 (iv) I ( ) d ) d ) = (I I ) d ) d ) = I b ) I (d d ) F N ( ) d ) d ) = (F N F N ) d ) d ) = F N b ) F N (d d ) for all b ) V, (d d ) V. Example.8. Cross product of SVNGSs Ǧb and Ǧb shown in Fig..0 is defined as Ǧb Ǧb = { Mathematics,07, 5, 60, } and is depicted in Fig of 0 b d 3(0.5,0.,0.6, 0.5, 0., 0.6) b 4d (0.,0.,0.6, 0., 0., 0.6) b d (0.,0.,0.8, 0., 0., 0.8) b d (0.3,0.,0.6, 0.3, 0., 0.6) (0.,0.,0.8, 0., 0., 0.8) b 4d 3(0.5,0.,0.6, 0.5, 0., 0.6) b 3d 3(0.4,0.,0.5, 0.4, 0., 0.5) (0.3,0.,0.6, 0.3, 0., 0.6) b 3d (0.3,0.,0.4, 0.3, 0., 0.4) (0.,0.,0.8, 0., 0., 0.8) b d (0.3,0.,0.8, 0.3, 0., 0.8) (0.3,0.,0.6, 0.3, 0., 0.6) b 4d (0.3,0.,0.6, 0.3, 0., 0.6) b d (0.,0.,0.6, 0., 0., 0.6) b d 3(0.5,0.,0.8, 0.5, 0., 0.8) b 3d (0.,0.,0.4, 0., 0., 0.4) Figure 3. Ǧ b Ǧ b. Figure.3: Ǧ b Ǧb Theorem. The cross product Ǧ b Ǧ b = (,,,..., m m ) of two SVNSGSs Theorem of GSs Gˇ s.9. and GˇCross s product Ǧb Ǧb = (,,,..., m m ) of two SVNSGSs of GSs Gˇ s and G ˇ is a SVNGS of Gˇ s Gˇ s. s is a SVNGS of Gˇ s G ˇ s. Proof. Proof. For Forb b b b V V,, dd dd V V, T ) ) T ( P (d d ) ) ( d ) d )) = T P b ) T P (d d ) [T ) T ] [T (d ) T P ) T P ] P (d ) T P = [T ) T (d )] [T ) T P ) T P (d )] P ( ) T P = T ) d ) T( P ) d ), ( P ) d ) P ) d ) T( N ) ( d ) d )) = T N b ) T N (d d ) T N ( [T N ) T N ] [T N (d ) T N ) ( d ) d )) = T N b ) T N (d d ) = [T [T N ) T (d )] [T N ) T N ) T N ] N (d ) T N (d )] = [T T( N ) d ) T( N ) T N (d )] [T N ) d ) ), T N = T N ( ) d ) T N ( ) d ) I( P ) ( d ) d )) = I P b ) I P (d d ) I [I P ) I P ] [I P (d ) I P ( P ) ( d ) d )) = I P = [I P b ) I ) I P P (d d ) (d )] [I P ) I P [I I( P P ) I P ] [I ) d ) I( P P (d ) I P ) d ), = [I P ) I P (d )] [I P ) I P = I( P 5 ) d ) I( P ) d ) I N ( ) ( d ) d )) = I N b ) I N (d d ) [I N ) I N ] [I N (d ) I N = [I N ) I N (d )] [I N ) I N = I N ( ) d ) I N ( ) d )

15 Mathematics 07, 5, 60 5 of 0 F( P ) ( d ) d )) = F P b ) F P (d d ) [F P ) F P ] [F P (d ) F P = [F P ) F P (d )] [F P ) F P = F( P ) d ) F( P ) d ) F N ( ) ( d ) d )) = F N b ) F N (d d ) where b d, b d V V and h {,,..., m}. [F N ) F N ] [F N (d ) F N = [F N ) F N (d )] [F N ) F N = F N ( ) d ) F N ( ) d ) Definition 7. Let Ǧ b = (,,,..., m ) and Ǧ b = (,,,..., m ) be two SVNGSs. The composition of Ǧ b and Ǧ b, denoted by Ǧ b Ǧ b = (,,,..., m m ) is defined as T( P ) d) = (TP T P )d) = T P ) T P (i) I( P ) d) = (IP I P )d) = I P ) I P F( P ) d) = (FP F P )d) = F P ) F P T N ( ) d) = (TN T N )d) = T N ) T N (ii) I N ( ) d) = (IN I N )d) = I N ) I N F N ( ) d) = (FN F N )d) = F N ) F N (iii) (iv) (v) (vi) (vii) for all d) V V, T( P ) d )d ) = (T P T P )d )d ) = T P ) T P (d d ) I( P ) d )d ) = (I P I P )d )d ) = I P ) I P (d d ) F( P ) d )d ) = (F P F P )d )d ) = F P ) F P (d d ) T N ( ) d )d ) = (T N T N )d )d ) = T N ) T N (d d ) I N ( ) d )d ) = (I N I N )d )d ) = I N ) I N (d d ) F N ( ) d )d ) = (F N F N )d )d ) = F N ) F N (d d ) for all b V, (d d ) V, T( P ) d) d) = (T P T P ) d) d) = T P T P b ) I( P ) d) d) = (I P I P ) d) d) = I P I P b ) F( P ) d) d) = (F P F P ) d) d) = F P F P b ) T N ( ) d) d) = (T N T N ) d) d) = T N T N b ) I N ( ) d) d) = (I N I N ) d) d) = I N I N b ) F N ( ) d) d) = (F N F N ) d) d) = F N F N b ) for all d V, b ) V, and T( P ) d ) d ) = (T P T P ) d ) d ) = T P b ) T P (d ) T P (d ) I( P ) d ) d ) = (I P I P ) d ) d ) = I P b ) I P (d ) I P (d ) F( P ) d ) d ) = (F P F P ) d ) d ) = F P b ) F P (d ) F P (d )

16 (vi) I ( ) d) d) = (I I ) d) d) = I I b ) F( N ) d) d) = (F N F N ) d) d) = F N F N b ) for all d V, b ) V. T( P ) d ) d ) = (T P T P ) d ) d ) = T P b ) T P (d ) T P (d ) (vii) Mathematics I 07, 5, 60 6 of 0 ( P ) d ) d ) = (I P I P ) d ) d ) = I P b ) I P (d ) I P (d ) F( P ) d ) d ) = (F P F P ) d ) d ) = F P b ) F P (d ) F P (d ) T N ( ) d ) d ) (T N T ) ) ) T (d ) T (d T( N I N ) d ) d ) = (T ( ) N d ) d ) (I N T I N ) d ) d ) = T N b ) T ) I N (d ) T (d ) I N (d ) (viii) I (d ) ( N F N ) d ) d ) = (I N ( I ) N ) d ) d ) = I F N b ) I N (d ) I b ) N N (d ) F (d ) F N ( N ) d ) d ) = (F N F N ) d ) d ) = F N b ) F N (d ) F N (d (d ) ) for forall all b b) ) V V,,(d (d d d )) VV such that d = d. Example.3. Composition of SVNGSs Example. The composition of SVNGSs Ǧb and Ǧ b and Ǧ Ǧb shown in Fig..0 is defined as b shown in Figure 0 is defined as Ǧ Ǧb b Ǧ Ǧb = b = {,, } and is depicted in Fig..4 and.5. {,, } and is depicted in Figures 4 and 5. (0.3,0.,0.8, 0.3, 0., 0.8) (0.,0.,0.6, 0., 0., 0.6) b d (0.,0.,0.6, 0., 0., 0.6) b d (0.3,0.,0.6, 0.3, 0., 0.6) (0.,0.,0.8, 0., 0., 0.8) (0.,0.,0.8, 0., 0., 0.8) b d 3(0.5,0.,0.8, 0.5, 0., 0.8) (0.,0.,0.8, 0., 0., 0.8) (0.5,0.,0.8, 0.5, 0., 0.8) b d 3(0.5,0.,0.6, 0.5, 0., 0.6) (0.,0.,0.8, 0., 0., 0.8) (0.3,0.,0.6, 0.3, 0., 0.6) (0.,0.,0.8, 0., 0., 0.8) (0.3,0.,0.8, 0.3, 0., 0.8) (0.3,0.,0.8, 0.3, 0., 0.8) b d (0.,0.,0.8, 0., 0., 0.8) (0.,0.,0.8, 0., 0., 0.8) (0.3,0.,0.8, 0.3, 0., 0.8) b d (0.3,0.,0.8, 0.3, 0., 0.8) (0.3,0.,0.5, 0.3, 0., 0.5) (0.,0.,0.6, 0., 0., 0.6) b 4d (0.,0.,0.6, 0., 0., 0.6) b 3d (0.3,0.,0.4, 0.3, 0., 0.4) Figure 4. Ǧ b Ǧ b. Figure.4: Ǧ b Ǧb (0.,0.,0.6, 0., 0., 0.6) (0.,0.,0.6, 0., 0., 0.6) b 3d 3(0.4,0.,0.5, 0.4, 0., 0.5) (0.,0.,0.6, 0., 0., 0.6) (0.4,0.,0.6, 0.4, 0., 0.6) b 4d 3(0.5,0.,0.6, 0.5, 0., 0.6) (0.3,0.,0.6, 0.3, 0., 0.6) (0.,0.,0.4, 0., 0., 0.4) 7 (0.3,0.,0.6, 0.3, 0., 0.6) (0.3,0.,0.6, 0.3, 0., 0.6) (0.,0.,0.6, 0., 0., 0.6) b 3d (0.,0.,0.4, 0., 0., 0.4) (0.,0.,0.6, 0., 0., 0.6) (0.3,0.,0.6, 0.3, 0., 0.6) b 4d (0.3,0.,0.6, 0.3, 0., 0.6) Figure 5. Ǧ b Ǧ b. Figure.5: Ǧ b Ǧb Theorem 3. The composition Ǧ b Ǧ b = (,,,..., m m ) of two SVNGSs of Theorem.3. Composition Ǧb Ǧb = (,,,..., m m ) of two SVNGSs GSs Gˇ of GSs s Gˇ and Gˇ s and s is G ˇ a SVNGS of Gˇ s is a SVNGS s Gˇ of s. s G ˇ s. Proof. Consider three cases: Case. For b V, d d V T( P ) (d )d )) = T P ) T P (d d ) T P ) [T P (d ) T P = [T P ) T P (d )] [T P ) T P P P

17 Mathematics 07, 5, 60 7 of 0 Case. For b V, d d V, T( P ) (d )d )) = T P ) T P (d d ) T P ) [T P (d ) T P = [T P ) T P (d )] [T P ) T P = T( P ) d ) T( P ) d ) T N ( ) (d )d )) = T N ) T N (d d ) T N ) [T N (d ) T N = [T N ) T N (d )] [T N ) T N = T N ( ) d ) T N ( ) d ) I( P ) (d )d )) = I P ) I P (d d ) I P ) [I P (d ) I P = [I P ) I P (d )] [I P ) I P = I( P ) d ) I( P ) d ) I N ( ) (d )d )) = I N ) I N (d d ) I N ) [I N (d ) I N = [I N ) I N (d )] [I N ) I N = I N ( ) d ) I N ( ) d ) F( P ) (d )d )) = F P ) F P (d d ) F P ) [F P (d ) F P = [F P ) F P (d )] [F P ) F P = F( P ) d ) F( P ) d ) F N ( ) (d )d )) = F N ) F N (d d ) F N ) [F N (d ) F N = [F N ) F N (d )] [F N ) F N = F N ( ) d ) F N ( ) d ) for bd, bd V V. Case. For b V, d d V, T P ( ) ((d b)(d b)) = T P ) T P (d d ) T P ) [T P (d ) T P = [T P ) T P (d )] [T P ) T P = T P ( ) (d b) T P ( ) (d b)

18 Mathematics 07, 5, 60 8 of 0 T N ( ) ((d b)(d b)) = T N ) T N (d d ) T N ) [T N (d ) T N = [T N ) T N (d )] [T N ) T N = T N ( ) (d b) T N ( ) (d b) I( P ) ((d b)(d b)) = I P ) I P (d d ) I P ) [I P (d ) I P = [I P ) I P (d )] [I P ) I P = I( P ) (d b) I( P ) (d b) I N ( ) ((d b)(d b)) = I N ) I N (d d ) I N ) [I N (d ) I N = [I N ) I N (d )] [I N ) I N = I N ( ) (d b) I N ( ) (d b) F( P ) ((d b)(d b)) = F P ) F P (d d ) F P ) [F P (d ) F P = [F P ) F P (d )] [F P ) F P = F( P ) (d b) F( P ) (d b) F N ( ) ((d b)(d b)) = F N ) F N (d d ) F N ) [F N (d ) F N = [F N ) F N (d )] [F N ) F N = F N ( ) (d b) F N ( ) (d b) for d b, d b V V. Case 3. For b ) V, (d d ) V such that d = d, T( P ) ( d ) d )) = T P b ) T P (d ) T P (d ) [T P ) T P ] [T P (d ) T P = [T P ) T P (d )] [T P ) T P = T( P ) d ) T( P ) d ) T N ( ) ( d ) d )) = T N b ) T N (d ) T N (d ) [T N ) T N ] [T N (d ) T N = [T N ) T N (d )] [T N ) T N = T N ( ) d ) T N ( ) d )

19 Mathematics 07, 5, 60 9 of 0 I( P ) ( d ) d )) = I P b ) I P (d ) I P (d ) [I P ) I P ] [I P (d ) I P = [I P ) I P (d )] [I P ) I P = I( P ) d ) I( P ) d ) I N ( ) ( d ) d )) = I N b ) I N (d ) I N (d ) [I N ) I N ] [I N (d ) I N = [I N ) I N (d )] [I N ) I N = I N ( ) d ) I N ( ) d ) F( P ) ( d ) d )) = F P b ) F P (d ) F P (d ) [F P ) F P ] [F P (d ) F P = [F P ) F P (d )] [F P ) F P = F( P ) d ) F( P ) d ) where b d, b d V V. F N ( ) ( d ) d )) = F N b ) F N (d ) F N (d ) All cases are satisfied for all {,,..., m}. 3. Conclusions [F N ) F N ] [F N (d ) F N = [F N ) F N (d )] [F N ) F N = F N ( ) d ) F N ( ) d ) The notion of bipolar fuzzy graphs is applicable in several domains of engineering, expert systems, pattern recognition, signal processing, neural networs, medical diagnosis and decision-maing. SVNGSs show more flexibility, compatibility and precision for a system than single-valued neutrosophic graph structures. In this research paper, we introduced certain concepts of SVNGSs and elaborated on them with suitable examples. Further, we defined some operations on SVNGSs and investigated some relevant properties of these operations. We intend to generalize our research of fuzzification to () concepts of SVN soft graph structures, () concepts of SVN rough fuzzy graph structures, (3) concepts of SVN fuzzy soft graph structures, and (4) concepts of SVN rough fuzzy soft graph structures. Acnowledgments: The authors are thanful to referees for their valuable comments and suggestion. Author Contributions: Muhammad Aram, Muzzamal Sitara and Florentin Smarandache conceived and designed the experiments; Muhammad Aram performed the experiments; Muhammad Aram and Muzzamal Sitara analyzed the data; Florentin Smarandache contributed reagents/materials/analysis tools; Muzzamal Sitara wrote the paper. Conflicts of Interest: The authors declare no conflict of interest regarding the publication of this research paper. References. Zadeh, L.A. Fuzzy sets. Inf. Control 965, 8, Atanassov, K. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 986, 0,

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